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Obviously ",{"__TSPROSE_proseElement":175,"schema":567,"data":568,"storageKey":554,"id":569},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":553},"inlinerMath-p3iSNTiCx-1",{"__TSPROSE_proseElement":175,"schema":571,"data":572},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," cm is absurdly large because it is bigger than the smaller side of the whole sheet, so the correct answer is ",{"__TSPROSE_proseElement":175,"schema":574,"data":575,"storageKey":423,"id":576},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-2",{"__TSPROSE_proseElement":175,"schema":578,"data":427},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-DvAOdJYip","what-s-in-the-box",{"__TSPROSE_proseElement":175,"schema":582,"data":583,"children":587,"id":861},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":584},{"title":585,"level":341,"attributes":586},"Standard Frames",[],[588,631,638,648,666],{"__TSPROSE_proseElement":175,"schema":589,"children":590},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[591,625],{"__TSPROSE_proseElement":175,"schema":592,"children":593,"id":624},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[594,597,603,606,612,615,621],{"__TSPROSE_proseElement":175,"schema":595,"data":596},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"After doomscrolling the news and catching the trend, you decide to launch a new business making sturdy steel frames. To reduce the weight and price, you decide the frame cross-section area should be ",{"__TSPROSE_proseElement":175,"schema":598,"data":599,"storageKey":601,"id":602},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":600},"120","$ 120 $","inlinerMath-YmZRSaRhs",{"__TSPROSE_proseElement":175,"schema":604,"data":605},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," square centimeters. The inner dimensions of the frame are ",{"__TSPROSE_proseElement":175,"schema":607,"data":608,"storageKey":610,"id":611},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":609},"4","$ 4 $","inlinerMath-ZXyaHjumL",{"__TSPROSE_proseElement":175,"schema":613,"data":614},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," cm in height and ",{"__TSPROSE_proseElement":175,"schema":616,"data":617,"storageKey":619,"id":620},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":618},"6","$ 6 $","inlinerMath-9rSWhNw25",{"__TSPROSE_proseElement":175,"schema":622,"data":623},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," cm in width. How thick should the frame be?","paragraph-Fvucx1Iw7",{"__TSPROSE_proseElement":175,"schema":626,"data":627,"storageKey":628,"id":630},{"name":385,"type":245,"linkable":175,"__TSPROSE_schema":175},{"src":628,"width":629},"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fsteel-frame.svg","500px","image-00x1QQ7Do",{"__TSPROSE_proseElement":175,"schema":632,"data":633},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":634,"serializedValidator":635},"Frame thickness",{"__ERUDIT_CHECK":175,"name":396,"data":636},{"expr":637},"3",{"__TSPROSE_proseElement":175,"schema":639,"children":640},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[641],{"__TSPROSE_proseElement":175,"schema":642,"children":643,"id":647},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[644],{"__TSPROSE_proseElement":175,"schema":645,"data":646},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"You can find the frame area using the area formula for a rectangle, but you have to add the unknown thickness twice to each side because the frame exists on both sides.","paragraph-z9oe5LQsZ",{"__TSPROSE_proseElement":175,"schema":649,"children":650},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[651],{"__TSPROSE_proseElement":175,"schema":652,"children":653,"id":665},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[654,657,662],{"__TSPROSE_proseElement":175,"schema":655,"data":656},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The frame thickness should be ",{"__TSPROSE_proseElement":175,"schema":658,"data":659,"storageKey":660,"id":661},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"$ 3 $","inlinerMath-mqn34wvB0",{"__TSPROSE_proseElement":175,"schema":663,"data":664},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," centimeters.","paragraph-zRs6hTAYD",{"__TSPROSE_proseElement":175,"schema":667,"children":668},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[669,701,707,721,727,734,740,754,760,774,780,825,831,854],{"__TSPROSE_proseElement":175,"schema":670,"children":671,"id":700},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[672,675,679,682,688,691,697],{"__TSPROSE_proseElement":175,"schema":673,"data":674},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"We find the frame area using the rectangle area formula, but we account for the frame thickness. Let the frame thickness be ",{"__TSPROSE_proseElement":175,"schema":676,"data":677,"storageKey":443,"id":678},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":442},"inlinerMath-TLSwLL8iH-2",{"__TSPROSE_proseElement":175,"schema":680,"data":681},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," centimeters. Then the full height becomes ",{"__TSPROSE_proseElement":175,"schema":683,"data":684,"storageKey":686,"id":687},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":685},"4 + 2x","$ 4 + 2x $","inlinerMath-OyHvIM5bm",{"__TSPROSE_proseElement":175,"schema":689,"data":690},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," centimeters, and the full width becomes ",{"__TSPROSE_proseElement":175,"schema":692,"data":693,"storageKey":695,"id":696},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":694},"6 + 2x","$ 6 + 2x $","inlinerMath-ywXbUWbXQ",{"__TSPROSE_proseElement":175,"schema":698,"data":699},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," centimeters. So the frame area is:","paragraph-RJKqrH82z",{"__TSPROSE_proseElement":175,"schema":702,"data":703,"storageKey":705,"id":706},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":704,"freeze":246},"S = (4 + 2x)(6 + 2x)","$$ S = (4 + 2x)(6 + 2x) $$","blockMath-bhiAbHUwl",{"__TSPROSE_proseElement":175,"schema":708,"children":709,"id":720},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[710,713,717],{"__TSPROSE_proseElement":175,"schema":711,"data":712},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Since the frame area must be ",{"__TSPROSE_proseElement":175,"schema":714,"data":715,"storageKey":601,"id":716},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":600},"inlinerMath-YmZRSaRhs-1",{"__TSPROSE_proseElement":175,"schema":718,"data":719},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," square centimeters, we write the equation:","paragraph-IClXOwBDE",{"__TSPROSE_proseElement":175,"schema":722,"data":723,"storageKey":725,"id":726},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":724,"freeze":246},"120 = (4 + 2x)(6 + 2x)","$$ 120 = (4 + 2x)(6 + 2x) $$","blockMath-zZ8sNN8WV",{"__TSPROSE_proseElement":175,"schema":728,"children":729,"id":733},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[730],{"__TSPROSE_proseElement":175,"schema":731,"data":732},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Expand the right-hand side and combine like terms:","paragraph-jhAAhx04P",{"__TSPROSE_proseElement":175,"schema":735,"data":736,"storageKey":738,"id":739},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":737,"freeze":246},"120 = 24 + 8x + 12x + 4x^2 \\\\ 120 = 24 + 20x + 4x^2","$$ 120 = 24 + 8x + 12x + 4x^2 \\\\ 120 = 24 + 20x + 4x^2 $$","blockMath-RdI6BLEkr",{"__TSPROSE_proseElement":175,"schema":741,"children":742,"id":753},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[743,746,750],{"__TSPROSE_proseElement":175,"schema":744,"data":745},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Subtract ",{"__TSPROSE_proseElement":175,"schema":747,"data":748,"storageKey":601,"id":749},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":600},"inlinerMath-YmZRSaRhs-2",{"__TSPROSE_proseElement":175,"schema":751,"data":752},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," from both sides:","paragraph-vBaIZzu03",{"__TSPROSE_proseElement":175,"schema":755,"data":756,"storageKey":758,"id":759},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":757,"freeze":246},"120 - 120 = 24 + 20x + 4x^2 - 120 \\\\ 0 = 4x^2 + 20x - 96","$$ 120 - 120 = 24 + 20x + 4x^2 - 120 \\\\ 0 = 4x^2 + 20x - 96 $$","blockMath-xnoC1JsaJ",{"__TSPROSE_proseElement":175,"schema":761,"children":762,"id":773},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[763,766,770],{"__TSPROSE_proseElement":175,"schema":764,"data":765},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Divide both sides by ",{"__TSPROSE_proseElement":175,"schema":767,"data":768,"storageKey":610,"id":769},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":609},"inlinerMath-ZXyaHjumL-1",{"__TSPROSE_proseElement":175,"schema":771,"data":772},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," to simplify the coefficients:","paragraph-AQThy4DbE",{"__TSPROSE_proseElement":175,"schema":775,"data":776,"storageKey":778,"id":779},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":777,"freeze":246},"0 = x^2 + 5x - 24","$$ 0 = x^2 + 5x - 24 $$","blockMath-rd6Yb504R",{"__TSPROSE_proseElement":175,"schema":781,"children":782,"id":824},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[783,785,789,792,798,800,806,808,814,816,822],{"__TSPROSE_proseElement":175,"schema":784,"data":490},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":786,"data":787,"storageKey":496,"id":788},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":495},"dependencyInliner-fdi2FzGId-1",{"__TSPROSE_proseElement":175,"schema":790,"data":791},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". We need two numbers whose sum is ",{"__TSPROSE_proseElement":175,"schema":793,"data":794,"storageKey":796,"id":797},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":795},"5","$ 5 $","inlinerMath-MAD3UaxRg",{"__TSPROSE_proseElement":175,"schema":799,"data":509},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":801,"data":802,"storageKey":804,"id":805},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":803},"-24","$ -24 $","inlinerMath-oYd9MM1ly",{"__TSPROSE_proseElement":175,"schema":807,"data":518},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":809,"data":810,"storageKey":812,"id":813},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":811},"8","$ 8 $","inlinerMath-fBsW71uIl",{"__TSPROSE_proseElement":175,"schema":815,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":817,"data":818,"storageKey":820,"id":821},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":819},"-3","$ -3 $","inlinerMath-nfmGVV06J",{"__TSPROSE_proseElement":175,"schema":823,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-eOBHgSXbq",{"__TSPROSE_proseElement":175,"schema":826,"data":827,"storageKey":829,"id":830},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":828,"freeze":246},"(x + 8)(x - 3) = 0","$$ (x + 8)(x - 3) = 0 $$","blockMath-hxkU1XQiR",{"__TSPROSE_proseElement":175,"schema":832,"children":833,"id":853},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[834,837,841,844,850],{"__TSPROSE_proseElement":175,"schema":835,"data":836},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"So the frame thickness must be ",{"__TSPROSE_proseElement":175,"schema":838,"data":839,"storageKey":660,"id":840},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"inlinerMath-mqn34wvB0-1",{"__TSPROSE_proseElement":175,"schema":842,"data":843},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," centimeters. The second root, ",{"__TSPROSE_proseElement":175,"schema":845,"data":846,"storageKey":848,"id":849},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":847},"-8","$ -8 $","inlinerMath-g4vfUaz94",{"__TSPROSE_proseElement":175,"schema":851,"data":852},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", is still a solution of the equation in pure math terms, but it does not work here because thickness cannot physically be negative.","paragraph-5bUsL7qE5",{"__TSPROSE_proseElement":175,"schema":855,"children":856,"id":860},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[857],{"__TSPROSE_proseElement":175,"schema":858,"data":859},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"This is a great example of how the real world puts restrictions on a problem while mathematics knows nothing about those restrictions. Once we translate a real-life situation into equations, we get \"clean\" mathematics, and it honestly gives every possible solution. But not all of them mean anything physically. That is why after solving an equation, you always need to check the answers against reality. 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Around it, workers need to lay a walkway of paving stones with the same width everywhere so lots of people can admire the thing at once. The total area of the installation together with the walkway must be twice the area of the model itself. Find the width of the walkway.","paragraph-eIF1O4pPc",{"__TSPROSE_proseElement":175,"schema":896,"data":897},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":898,"serializedValidator":899},"Walkway width",{"__ERUDIT_CHECK":175,"name":396,"data":900},{"expr":398},{"__TSPROSE_proseElement":175,"schema":902,"children":903},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[904],{"__TSPROSE_proseElement":175,"schema":905,"children":906,"id":926},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[907,910,914,917,923],{"__TSPROSE_proseElement":175,"schema":908,"data":909},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"If the walkway width is ",{"__TSPROSE_proseElement":175,"schema":911,"data":912,"storageKey":443,"id":913},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":442},"inlinerMath-TLSwLL8iH-3",{"__TSPROSE_proseElement":175,"schema":915,"data":916},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", then each side of the installation increases by ",{"__TSPROSE_proseElement":175,"schema":918,"data":919,"storageKey":921,"id":922},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":920},"2x","$ 2x $","inlinerMath-pJu8ntKOb",{"__TSPROSE_proseElement":175,"schema":924,"data":925},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", since the walkway appears on both sides of the sculpture.","paragraph-rv84HJEEl",{"__TSPROSE_proseElement":175,"schema":928,"children":929},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[930],{"__TSPROSE_proseElement":175,"schema":931,"children":932,"id":943},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[933,936,940],{"__TSPROSE_proseElement":175,"schema":934,"data":935},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The walkway width is ",{"__TSPROSE_proseElement":175,"schema":937,"data":938,"storageKey":423,"id":939},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-3",{"__TSPROSE_proseElement":175,"schema":941,"data":942},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meters.","paragraph-oYpPasHPy",{"__TSPROSE_proseElement":175,"schema":945,"children":946},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[947,984,1009,1015,1022,1028,1041,1047,1098,1104],{"__TSPROSE_proseElement":175,"schema":948,"children":949,"id":983},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[950,953,957,960,964,967,973,975,981],{"__TSPROSE_proseElement":175,"schema":951,"data":952},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Let the walkway width be ",{"__TSPROSE_proseElement":175,"schema":954,"data":955,"storageKey":443,"id":956},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":442},"inlinerMath-TLSwLL8iH-4",{"__TSPROSE_proseElement":175,"schema":958,"data":959},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". The walkway surrounds the sculpture from all sides, so each side grows by ",{"__TSPROSE_proseElement":175,"schema":961,"data":962,"storageKey":443,"id":963},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":442},"inlinerMath-TLSwLL8iH-5",{"__TSPROSE_proseElement":175,"schema":965,"data":966},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," on both ends. That means the total dimensions become ",{"__TSPROSE_proseElement":175,"schema":968,"data":969,"storageKey":971,"id":972},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":970},"(8 + 2x)","$ (8 + 2x) $","inlinerMath-5P001hOIF",{"__TSPROSE_proseElement":175,"schema":974,"data":884},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":976,"data":977,"storageKey":979,"id":980},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":978},"(12 + 2x)","$ (12 + 2x) $","inlinerMath-qAB3geJPK",{"__TSPROSE_proseElement":175,"schema":982,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-HdiC5lckd",{"__TSPROSE_proseElement":175,"schema":985,"children":986,"id":1008},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[987,990,996,999,1005],{"__TSPROSE_proseElement":175,"schema":988,"data":989},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The area of the sculpture itself is ",{"__TSPROSE_proseElement":175,"schema":991,"data":992,"storageKey":994,"id":995},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":993},"8 \\cdot 12 = 96","$ 8 \\cdot 12 = 96 $","inlinerMath-aSrzc6HDW",{"__TSPROSE_proseElement":175,"schema":997,"data":998},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," square meters. The problem says the final area must be twice that, so ",{"__TSPROSE_proseElement":175,"schema":1000,"data":1001,"storageKey":1003,"id":1004},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1002},"2 \\cdot 96 = 192","$ 2 \\cdot 96 = 192 $","inlinerMath-rtEG77iWj",{"__TSPROSE_proseElement":175,"schema":1006,"data":1007},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," square meters. Write the equation:","paragraph-3v0Ctrs4f",{"__TSPROSE_proseElement":175,"schema":1010,"data":1011,"storageKey":1013,"id":1014},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1012,"freeze":246},"(8 + 2x)(12 + 2x) = 192","$$ (8 + 2x)(12 + 2x) = 192 $$","blockMath-tnAgjLAu7",{"__TSPROSE_proseElement":175,"schema":1016,"children":1017,"id":1021},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1018],{"__TSPROSE_proseElement":175,"schema":1019,"data":1020},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Expand and simplify:","paragraph-AUzV4S1Ts",{"__TSPROSE_proseElement":175,"schema":1023,"data":1024,"storageKey":1026,"id":1027},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1025,"freeze":246},"96 + 16x + 24x + 4x^2 = 192 \\\\ 4x^2 + 40x + 96 = 192 \\\\ 4x^2 + 40x - 96 = 0","$$ 96 + 16x + 24x + 4x^2 = 192 \\\\ 4x^2 + 40x + 96 = 192 \\\\ 4x^2 + 40x - 96 = 0 $$","blockMath-5doyvVsYD",{"__TSPROSE_proseElement":175,"schema":1029,"children":1030,"id":1040},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1031,1033,1037],{"__TSPROSE_proseElement":175,"schema":1032,"data":765},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":1034,"data":1035,"storageKey":610,"id":1036},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":609},"inlinerMath-ZXyaHjumL-2",{"__TSPROSE_proseElement":175,"schema":1038,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},":","paragraph-HVyY5904F",{"__TSPROSE_proseElement":175,"schema":1042,"data":1043,"storageKey":1045,"id":1046},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1044,"freeze":246},"x^2 + 10x - 24 = 0","$$ x^2 + 10x - 24 = 0 $$","blockMath-7tiOl5mWj",{"__TSPROSE_proseElement":175,"schema":1048,"children":1049,"id":1097},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1050,1052,1056,1059,1065,1068,1072,1074,1078,1081,1087,1090,1094],{"__TSPROSE_proseElement":175,"schema":1051,"data":490},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":1053,"data":1054,"storageKey":496,"id":1055},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":495},"dependencyInliner-fdi2FzGId-2",{"__TSPROSE_proseElement":175,"schema":1057,"data":1058},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". 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Let ",{"__TSPROSE_proseElement":175,"schema":1274,"data":1275,"storageKey":1277,"id":1278},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1276},"P","$ P $","inlinerMath-NNMK5da4D",{"__TSPROSE_proseElement":175,"schema":1280,"data":1281},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," be the package price for the first class. Then the cheaper packages bought by the second class cost ",{"__TSPROSE_proseElement":175,"schema":1283,"data":1284,"storageKey":1286,"id":1287},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1285},"P - 20","$ P - 20 $","inlinerMath-EYpPgWjnc",{"__TSPROSE_proseElement":175,"schema":1289,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-hroazl0e9",{"__TSPROSE_proseElement":175,"schema":1292,"data":1293,"storageKey":1295,"id":1296},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1294,"freeze":246},"\\frac{840}{P - 20} = \\frac{840}{P} + 1","$$ \\frac{840}{P - 20} = \\frac{840}{P} + 1 $$","blockMath-FzkWP9e6Z",{"__TSPROSE_proseElement":175,"schema":1298,"children":1299,"id":1311},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1300,1303,1309],{"__TSPROSE_proseElement":175,"schema":1301,"data":1302},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"We got an equation with one variable. Let's solve it. First multiply both sides by ",{"__TSPROSE_proseElement":175,"schema":1304,"data":1305,"storageKey":1307,"id":1308},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1306},"P(P - 20)","$ P(P - 20) $","inlinerMath-x7EmHHa2j",{"__TSPROSE_proseElement":175,"schema":1310,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-fH4Arlko5",{"__TSPROSE_proseElement":175,"schema":1313,"data":1314,"storageKey":1316,"id":1317},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1315,"freeze":246},"840P = 840(P - 20) + P(P - 20)","$$ 840P = 840(P - 20) + P(P - 20) $$","blockMath-578udbem3",{"__TSPROSE_proseElement":175,"schema":1319,"children":1320,"id":1324},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1321],{"__TSPROSE_proseElement":175,"schema":1322,"data":1323},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Simplify:","paragraph-LpW8VvKO3",{"__TSPROSE_proseElement":175,"schema":1326,"data":1327,"storageKey":1329,"id":1330},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1328,"freeze":246},"\\cancel{840P} = \\cancel{840P} - 20 \\cdot 840 + P^2 - 20P \\\\ 0 = - 20 \\cdot 840 + P^2 - 20P \\\\ 0 = P^2 - 20P - 20 \\cdot 840","$$ \\cancel{840P} = \\cancel{840P} - 20 \\cdot 840 + P^2 - 20P \\\\ 0 = - 20 \\cdot 840 + P^2 - 20P \\\\ 0 = P^2 - 20P - 20 \\cdot 840 $$","blockMath-oc9foEtBa",{"__TSPROSE_proseElement":175,"schema":1332,"children":1333,"id":1354},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1334,1337,1343,1346,1352],{"__TSPROSE_proseElement":175,"schema":1335,"data":1336},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Once again coefficient ",{"__TSPROSE_proseElement":175,"schema":1338,"data":1339,"storageKey":1341,"id":1342},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1340},"B","$ B $","inlinerMath-RvASDBkSH",{"__TSPROSE_proseElement":175,"schema":1344,"data":1345},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is even, so we can use the ",{"__TSPROSE_proseElement":175,"schema":1347,"data":1348,"storageKey":1350,"id":1351},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":1349},"simplified quadratic formula","\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002Fpractice\u002F$evenB","dependencyInliner-pF48j3mqL",{"__TSPROSE_proseElement":175,"schema":1353,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-hpK6lvEmb",{"__TSPROSE_proseElement":175,"schema":1356,"data":1357,"storageKey":1359,"id":1360},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1358,"freeze":246},"P = -(-10) \\pm \\sqrt{(-10)^2 - (-20\\cdot 840)} = 10 \\pm \\sqrt{100 + 2 \\cdot 84 \\cdot 100} = 10 \\pm \\sqrt{100(1 + 168)} = 10 \\pm \\sqrt{100}\\sqrt{169} = 10 \\pm 130","$$ P = -(-10) \\pm \\sqrt{(-10)^2 - (-20\\cdot 840)} = 10 \\pm \\sqrt{100 + 2 \\cdot 84 \\cdot 100} = 10 \\pm \\sqrt{100(1 + 168)} = 10 \\pm \\sqrt{100}\\sqrt{169} = 10 \\pm 130 $$","blockMath-qyfZge33u",{"__TSPROSE_proseElement":175,"schema":1362,"children":1363,"id":1391},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1364,1367,1373,1375,1381,1384,1388],{"__TSPROSE_proseElement":175,"schema":1365,"data":1366},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The possible prices are ",{"__TSPROSE_proseElement":175,"schema":1368,"data":1369,"storageKey":1371,"id":1372},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1370},"P = 140","$ P = 140 $","inlinerMath-zbwSFrq4X",{"__TSPROSE_proseElement":175,"schema":1374,"data":1118},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":1376,"data":1377,"storageKey":1379,"id":1380},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1378},"P = -120","$ P = -120 $","inlinerMath-wMXDueRPg",{"__TSPROSE_proseElement":175,"schema":1382,"data":1383},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Negative price is bullshit, so ",{"__TSPROSE_proseElement":175,"schema":1385,"data":1386,"storageKey":1371,"id":1387},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1370},"inlinerMath-zbwSFrq4X-1",{"__TSPROSE_proseElement":175,"schema":1389,"data":1390},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". That is the more expensive package. Now find how many packages the first class bought:","paragraph-SOdh6itJG",{"__TSPROSE_proseElement":175,"schema":1393,"data":1394,"storageKey":1396,"id":1397},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1395,"freeze":246},"N_1 = \\frac{840}{140} = 6","$$ N_1 = \\frac{840}{140} = 6 $$","blockMath-SWlvkefPA",{"__TSPROSE_proseElement":175,"schema":1399,"children":1400,"id":1412},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1401,1404,1410],{"__TSPROSE_proseElement":175,"schema":1402,"data":1403},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The second class bought one more package, so ",{"__TSPROSE_proseElement":175,"schema":1405,"data":1406,"storageKey":1408,"id":1409},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1407},"N_2 = 7","$ N_2 = 7 $","inlinerMath-wiTlWGPh0",{"__TSPROSE_proseElement":175,"schema":1411,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-FnwWz344D","buying-cookies",{"__TSPROSE_proseElement":175,"schema":1415,"data":1416,"children":1421,"id":1757},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":1417},{"title":1418,"level":1419,"attributes":1420},"Age Problems...","hard",[],[1422,1432,1444,1454,1464,1489],{"__TSPROSE_proseElement":175,"schema":1423,"children":1424},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[1425],{"__TSPROSE_proseElement":175,"schema":1426,"children":1427,"id":1431},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1428],{"__TSPROSE_proseElement":175,"schema":1429,"data":1430},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Six years ago, Paul's age was equal to twice the square of Anna's age. In four years, Anna's age will be one fourth of Paul's age. How old are Paul and Anna now?","paragraph-ttc8VFOCr",{"__TSPROSE_proseElement":175,"schema":1433,"data":1434},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":1435,"serializedValidator":1436},"Paul and Anna's ages",{"type":1177,"ordered":246,"separator":1178,"answers":1437},[1438,1441],{"__ERUDIT_CHECK":175,"name":396,"data":1439},{"expr":1440},"56",{"__ERUDIT_CHECK":175,"name":396,"data":1442},{"expr":1443},"11",{"__TSPROSE_proseElement":175,"schema":1445,"children":1446},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[1447],{"__TSPROSE_proseElement":175,"schema":1448,"children":1449,"id":1453},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1450],{"__TSPROSE_proseElement":175,"schema":1451,"data":1452},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"When you set up equations about the past or the future, remember that both ages differ from the current ages by however many years you move backward or forward.","paragraph-CCJ3oSnXm",{"__TSPROSE_proseElement":175,"schema":1455,"children":1456},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[1457],{"__TSPROSE_proseElement":175,"schema":1458,"children":1459,"id":1463},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1460],{"__TSPROSE_proseElement":175,"schema":1461,"data":1462},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"This is a real-world problem, so in both situations the ages of Paul and Anna must be positive numbers.","paragraph-HJwoH9o89",{"__TSPROSE_proseElement":175,"schema":1465,"children":1466},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[1467],{"__TSPROSE_proseElement":175,"schema":1468,"children":1469,"id":1488},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1470,1473,1478,1481,1486],{"__TSPROSE_proseElement":175,"schema":1471,"data":1472},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Paul is now ",{"__TSPROSE_proseElement":175,"schema":1474,"data":1475,"storageKey":1476,"id":1477},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1440},"$ 56 $","inlinerMath-WYRLesvjE",{"__TSPROSE_proseElement":175,"schema":1479,"data":1480},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," years old, and Anna is ",{"__TSPROSE_proseElement":175,"schema":1482,"data":1483,"storageKey":1484,"id":1485},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1443},"$ 11 $","inlinerMath-vFI6JrYG1",{"__TSPROSE_proseElement":175,"schema":1487,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-smZns45Pg",{"__TSPROSE_proseElement":175,"schema":1490,"children":1491},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[1492,1517,1523,1530,1536,1558,1564,1586,1592,1606,1612,1681,1687,1731,1737],{"__TSPROSE_proseElement":175,"schema":1493,"children":1494,"id":1516},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1495,1498,1504,1507,1513],{"__TSPROSE_proseElement":175,"schema":1496,"data":1497},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Let Paul's age be ",{"__TSPROSE_proseElement":175,"schema":1499,"data":1500,"storageKey":1502,"id":1503},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1501},"p","$ p $","inlinerMath-Dva0fpGoR",{"__TSPROSE_proseElement":175,"schema":1505,"data":1506},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and Anna's age be ",{"__TSPROSE_proseElement":175,"schema":1508,"data":1509,"storageKey":1511,"id":1512},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1510},"a","$ a $","inlinerMath-YqDfaaRXA",{"__TSPROSE_proseElement":175,"schema":1514,"data":1515},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Translate the first sentence into an equation, keeping in mind that six years must be subtracted from both ages:","paragraph-Vjs2dVayr",{"__TSPROSE_proseElement":175,"schema":1518,"data":1519,"storageKey":1521,"id":1522},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1520,"freeze":246},"p-6 = 2(a-6)^2","$$ p-6 = 2(a-6)^2 $$","blockMath-I41LAa9Ra",{"__TSPROSE_proseElement":175,"schema":1524,"children":1525,"id":1529},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1526],{"__TSPROSE_proseElement":175,"schema":1527,"data":1528},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Now write the equation for the second sentence:","paragraph-1pIYTRS6d",{"__TSPROSE_proseElement":175,"schema":1531,"data":1532,"storageKey":1534,"id":1535},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1533,"freeze":246},"a + 4 = \\frac{1}{4}(p + 4) \\\\ 4a + 16 = p + 4","$$ a + 4 = \\frac{1}{4}(p + 4) \\\\ 4a + 16 = p + 4 $$","blockMath-MiV7IEH9y",{"__TSPROSE_proseElement":175,"schema":1537,"children":1538,"id":1557},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1539,1542,1548,1551,1555],{"__TSPROSE_proseElement":175,"schema":1540,"data":1541},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Actually, let's reshape that equation a bit so the right side becomes ",{"__TSPROSE_proseElement":175,"schema":1543,"data":1544,"storageKey":1546,"id":1547},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1545},"p - 6","$ p - 6 $","inlinerMath-sSpVYa7JE",{"__TSPROSE_proseElement":175,"schema":1549,"data":1550},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", which we can substitute into the first one. Subtract ",{"__TSPROSE_proseElement":175,"schema":1552,"data":1553,"storageKey":1085,"id":1554},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1084},"inlinerMath-8Y3Z60Vbg-1",{"__TSPROSE_proseElement":175,"schema":1556,"data":752},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-NNkMdvQat",{"__TSPROSE_proseElement":175,"schema":1559,"data":1560,"storageKey":1562,"id":1563},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1561,"freeze":246},"4a + 16 - 10 = p + 4 - 10 \\\\ 4a + 6 = p - 6","$$ 4a + 16 - 10 = p + 4 - 10 \\\\ 4a + 6 = p - 6 $$","blockMath-QqpHA0n76",{"__TSPROSE_proseElement":175,"schema":1565,"children":1566,"id":1585},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1567,1570,1574,1577,1583],{"__TSPROSE_proseElement":175,"schema":1568,"data":1569},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"In the first equation, replace ",{"__TSPROSE_proseElement":175,"schema":1571,"data":1572,"storageKey":1546,"id":1573},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1545},"inlinerMath-sSpVYa7JE-1",{"__TSPROSE_proseElement":175,"schema":1575,"data":1576},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," with ",{"__TSPROSE_proseElement":175,"schema":1578,"data":1579,"storageKey":1581,"id":1582},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1580},"4a + 6","$ 4a + 6 $","inlinerMath-Pjy0t3MYD",{"__TSPROSE_proseElement":175,"schema":1584,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-kQJhmKZHY",{"__TSPROSE_proseElement":175,"schema":1587,"data":1588,"storageKey":1590,"id":1591},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1589,"freeze":246},"4a + 6 = 2(a - 6)^2 \\\\ 4a + 6 = 2(a^2 - 12a + 36) \\\\ 4a + 6 = 2a^2 - 24a + 72 \\\\ 0 = 2a^2 - 28a + 66 \\\\","$$ 4a + 6 = 2(a - 6)^2 \\\\ 4a + 6 = 2(a^2 - 12a + 36) \\\\ 4a + 6 = 2a^2 - 24a + 72 \\\\ 0 = 2a^2 - 28a + 66 \\\\ $$","blockMath-aktZSQNvK",{"__TSPROSE_proseElement":175,"schema":1593,"children":1594,"id":1605},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1595,1598,1602],{"__TSPROSE_proseElement":175,"schema":1596,"data":1597},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"We got a quadratic equation. Divide both sides by ",{"__TSPROSE_proseElement":175,"schema":1599,"data":1600,"storageKey":423,"id":1601},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-5",{"__TSPROSE_proseElement":175,"schema":1603,"data":1604},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," to simplify it:","paragraph-myNpKkHde",{"__TSPROSE_proseElement":175,"schema":1607,"data":1608,"storageKey":1610,"id":1611},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1609,"freeze":246},"0 = a^2 - 14a + 33","$$ 0 = a^2 - 14a + 33 $$","blockMath-PLouTXYsj",{"__TSPROSE_proseElement":175,"schema":1613,"children":1614,"id":1680},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1615,1618,1624,1627,1633,1636,1640,1643,1647,1650,1656,1658,1662,1665,1671,1674,1678],{"__TSPROSE_proseElement":175,"schema":1616,"data":1617},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Coefficient ",{"__TSPROSE_proseElement":175,"schema":1619,"data":1620,"storageKey":1622,"id":1623},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1621},"A","$ A $","inlinerMath-eIvv0Olnw",{"__TSPROSE_proseElement":175,"schema":1625,"data":1626},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is ",{"__TSPROSE_proseElement":175,"schema":1628,"data":1629,"storageKey":1631,"id":1632},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1630},"1","$ 1 $","inlinerMath-GWJ3eIxvK",{"__TSPROSE_proseElement":175,"schema":1634,"data":1635},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", and the other numbers are simple. So try manual factoring. Since ",{"__TSPROSE_proseElement":175,"schema":1637,"data":1638,"storageKey":1341,"id":1639},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1340},"inlinerMath-RvASDBkSH-1",{"__TSPROSE_proseElement":175,"schema":1641,"data":1642},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is negative and ",{"__TSPROSE_proseElement":175,"schema":1644,"data":1645,"storageKey":1063,"id":1646},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1062},"inlinerMath-EcPvQ2j3c-1",{"__TSPROSE_proseElement":175,"schema":1648,"data":1649},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is positive, both numbers should be negative. Those numbers are ",{"__TSPROSE_proseElement":175,"schema":1651,"data":1652,"storageKey":1654,"id":1655},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1653},"-11","$ -11 $","inlinerMath-kIVTwwPZG",{"__TSPROSE_proseElement":175,"schema":1657,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":1659,"data":1660,"storageKey":820,"id":1661},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":819},"inlinerMath-nfmGVV06J-1",{"__TSPROSE_proseElement":175,"schema":1663,"data":1664},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," because their sum is ",{"__TSPROSE_proseElement":175,"schema":1666,"data":1667,"storageKey":1669,"id":1670},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1668},"-14","$ -14 $","inlinerMath-7JAwVNMQL",{"__TSPROSE_proseElement":175,"schema":1672,"data":1673},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and their product is ",{"__TSPROSE_proseElement":175,"schema":1675,"data":1676,"storageKey":554,"id":1677},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":553},"inlinerMath-p3iSNTiCx-2",{"__TSPROSE_proseElement":175,"schema":1679,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-3PmWhNGId",{"__TSPROSE_proseElement":175,"schema":1682,"data":1683,"storageKey":1685,"id":1686},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1684,"freeze":246},"(a - 11)(a - 3) = 0","$$ (a - 11)(a - 3) = 0 $$","blockMath-FXow7j64F",{"__TSPROSE_proseElement":175,"schema":1688,"children":1689,"id":1730},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1690,1693,1697,1700,1704,1707,1711,1714,1718,1721,1727],{"__TSPROSE_proseElement":175,"schema":1691,"data":1692},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"So Anna could be ",{"__TSPROSE_proseElement":175,"schema":1694,"data":1695,"storageKey":1484,"id":1696},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1443},"inlinerMath-vFI6JrYG1-1",{"__TSPROSE_proseElement":175,"schema":1698,"data":1699},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," years old or ",{"__TSPROSE_proseElement":175,"schema":1701,"data":1702,"storageKey":660,"id":1703},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"inlinerMath-mqn34wvB0-2",{"__TSPROSE_proseElement":175,"schema":1705,"data":1706},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," years old. She cannot be ",{"__TSPROSE_proseElement":175,"schema":1708,"data":1709,"storageKey":660,"id":1710},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"inlinerMath-mqn34wvB0-3",{"__TSPROSE_proseElement":175,"schema":1712,"data":1713},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", because then six years ago she would not even exist and the whole problem would be idiotic. So the only option left is that Anna is now ",{"__TSPROSE_proseElement":175,"schema":1715,"data":1716,"storageKey":1484,"id":1717},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1443},"inlinerMath-vFI6JrYG1-2",{"__TSPROSE_proseElement":175,"schema":1719,"data":1720},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Plug that into any convenient equation above, for example ",{"__TSPROSE_proseElement":175,"schema":1722,"data":1723,"storageKey":1725,"id":1726},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1724},"4a + 6 = p - 6","$ 4a + 6 = p - 6 $","inlinerMath-JdI8TDC4R",{"__TSPROSE_proseElement":175,"schema":1728,"data":1729},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", and find Paul's age:","paragraph-6QrJM0ThE",{"__TSPROSE_proseElement":175,"schema":1732,"data":1733,"storageKey":1735,"id":1736},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1734,"freeze":246},"4 \\cdot 11 + 6 = p - 6 \\\\ 44 + 6 = p - 6 \\\\ 50 = p - 6 \\\\ p = 56","$$ 4 \\cdot 11 + 6 = p - 6 \\\\ 44 + 6 = p - 6 \\\\ 50 = p - 6 \\\\ p = 56 $$","blockMath-eIlP2obVQ",{"__TSPROSE_proseElement":175,"schema":1738,"children":1739,"id":1756},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1740,1743,1747,1750,1754],{"__TSPROSE_proseElement":175,"schema":1741,"data":1742},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"So Paul is ",{"__TSPROSE_proseElement":175,"schema":1744,"data":1745,"storageKey":1476,"id":1746},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1440},"inlinerMath-WYRLesvjE-1",{"__TSPROSE_proseElement":175,"schema":1748,"data":1749},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," years old and Anna is ",{"__TSPROSE_proseElement":175,"schema":1751,"data":1752,"storageKey":1484,"id":1753},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1443},"inlinerMath-vFI6JrYG1-3",{"__TSPROSE_proseElement":175,"schema":1755,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-LfQqLXMmj","age-problems",{"__TSPROSE_proseElement":175,"schema":1759,"data":1760,"children":1764,"id":2147},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":1761},{"title":1762,"level":1419,"attributes":1763},"The Perfectionist's Garden",[],[1765,1807,1817,1838,1861],{"__TSPROSE_proseElement":175,"schema":1766,"children":1767},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[1768],{"__TSPROSE_proseElement":175,"schema":1769,"children":1770,"id":1806},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1771,1774,1778,1781,1787,1790,1796,1799,1803],{"__TSPROSE_proseElement":175,"schema":1772,"data":1773},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"In a city park there is a large rectangular flower bed measuring ",{"__TSPROSE_proseElement":175,"schema":1775,"data":1776,"storageKey":1085,"id":1777},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1084},"inlinerMath-8Y3Z60Vbg-2",{"__TSPROSE_proseElement":175,"schema":1779,"data":1780},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meters by ",{"__TSPROSE_proseElement":175,"schema":1782,"data":1783,"storageKey":1785,"id":1786},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1784},"15","$ 15 $","inlinerMath-1xC295UiR",{"__TSPROSE_proseElement":175,"schema":1788,"data":1789},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meters. The park owner, being a perfectionist, wants to plant ",{"__TSPROSE_proseElement":175,"schema":1791,"data":1792,"storageKey":1794,"id":1795},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1793},"36","$ 36 $","inlinerMath-84Rv0uj32",{"__TSPROSE_proseElement":175,"schema":1797,"data":1798},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," flowers on this bed so that the equal spacing between the flowers along the longer side is ",{"__TSPROSE_proseElement":175,"schema":1800,"data":1801,"storageKey":1631,"id":1802},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1630},"inlinerMath-GWJ3eIxvK-1",{"__TSPROSE_proseElement":175,"schema":1804,"data":1805},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meter greater than along the shorter side. Find the spacing between the flowers along each side.","paragraph-DtOwBF6Cp",{"__TSPROSE_proseElement":175,"schema":1808,"data":1809},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":1810,"serializedValidator":1811},"Spacing between flowers",{"type":1177,"ordered":246,"separator":1178,"answers":1812},[1813,1815],{"__ERUDIT_CHECK":175,"name":396,"data":1814},{"expr":398},{"__ERUDIT_CHECK":175,"name":396,"data":1816},{"expr":637},{"__TSPROSE_proseElement":175,"schema":1818,"children":1819},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[1820],{"__TSPROSE_proseElement":175,"schema":1821,"children":1822,"id":1837},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1823,1826,1834],{"__TSPROSE_proseElement":175,"schema":1824,"data":1825},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"To find the spacing between flowers, divide the side length by the number of flowers ",{"__TSPROSE_proseElement":175,"schema":1827,"data":1828,"children":1829,"id":1833},{"name":274,"type":255,"linkable":175,"__TSPROSE_schema":175},{"type":276},[1830],{"__TSPROSE_proseElement":175,"schema":1831,"data":1832},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"minus one!","emphasis-V1IoRBySj",{"__TSPROSE_proseElement":175,"schema":1835,"data":1836},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," The number of gaps between flowers on a side is always one less than the number of flowers planted along that side.","paragraph-BlQUplo27",{"__TSPROSE_proseElement":175,"schema":1839,"children":1840},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[1841],{"__TSPROSE_proseElement":175,"schema":1842,"children":1843,"id":1860},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1844,1847,1851,1854,1858],{"__TSPROSE_proseElement":175,"schema":1845,"data":1846},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The spacing along the longer side is ",{"__TSPROSE_proseElement":175,"schema":1848,"data":1849,"storageKey":660,"id":1850},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"inlinerMath-mqn34wvB0-4",{"__TSPROSE_proseElement":175,"schema":1852,"data":1853},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meters, and along the shorter side it is ",{"__TSPROSE_proseElement":175,"schema":1855,"data":1856,"storageKey":423,"id":1857},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-6",{"__TSPROSE_proseElement":175,"schema":1859,"data":942},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-6wlvKK1Zq",{"__TSPROSE_proseElement":175,"schema":1862,"children":1863},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[1864,1895,1902,1908,1915,1936,1942,1979,1985,2001,2007,2013,2019,2031,2037,2058,2064,2095,2101,2115,2121],{"__TSPROSE_proseElement":175,"schema":1865,"children":1866,"id":1894},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1867,1870,1874,1877,1883,1886,1892],{"__TSPROSE_proseElement":175,"schema":1868,"data":1869},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Let ",{"__TSPROSE_proseElement":175,"schema":1871,"data":1872,"storageKey":443,"id":1873},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":442},"inlinerMath-TLSwLL8iH-6",{"__TSPROSE_proseElement":175,"schema":1875,"data":1876},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," be the number of flowers along the longer side and ",{"__TSPROSE_proseElement":175,"schema":1878,"data":1879,"storageKey":1881,"id":1882},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1880},"y","$ y $","inlinerMath-GydTYC7Nc",{"__TSPROSE_proseElement":175,"schema":1884,"data":1885},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," the number along the shorter side. The product of these numbers must equal the total number of flowers, so ",{"__TSPROSE_proseElement":175,"schema":1887,"data":1888,"storageKey":1890,"id":1891},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1889},"xy = 36","$ xy = 36 $","inlinerMath-eyJl4CLbq",{"__TSPROSE_proseElement":175,"schema":1893,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-DZeAjpeSn",{"__TSPROSE_proseElement":175,"schema":1896,"children":1897,"id":1901},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1898],{"__TSPROSE_proseElement":175,"schema":1899,"data":1900},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Now calculate the spacing along each side:","paragraph-zfaaZPSgX",{"__TSPROSE_proseElement":175,"schema":1903,"data":1904,"storageKey":1906,"id":1907},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1905,"freeze":246},"\\frac{15}{x-1} >>{big} \\frac{10}{y-1}","$$ \\frac{15}{x-1} >>{big} \\frac{10}{y-1} $$","blockMath-1NIIXTAOx",{"__TSPROSE_proseElement":175,"schema":1909,"children":1910,"id":1914},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1911],{"__TSPROSE_proseElement":175,"schema":1912,"data":1913},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"We subtract one because the number of gaps between flowers on a side is always one less than the number of flowers along that side. Two flowers have just one gap between them, for example.","paragraph-kSfdYGDrS",{"__TSPROSE_proseElement":175,"schema":1916,"children":1917,"id":1935},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1918,1921,1925,1928,1932],{"__TSPROSE_proseElement":175,"schema":1919,"data":1920},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The problem says the larger spacing, the one with ",{"__TSPROSE_proseElement":175,"schema":1922,"data":1923,"storageKey":443,"id":1924},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":442},"inlinerMath-TLSwLL8iH-7",{"__TSPROSE_proseElement":175,"schema":1926,"data":1927},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", is ",{"__TSPROSE_proseElement":175,"schema":1929,"data":1930,"storageKey":1631,"id":1931},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1630},"inlinerMath-GWJ3eIxvK-2",{"__TSPROSE_proseElement":175,"schema":1933,"data":1934},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meter greater than the smaller one. So write:","paragraph-9tiCFlmaX",{"__TSPROSE_proseElement":175,"schema":1937,"data":1938,"storageKey":1940,"id":1941},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":1939,"freeze":246},"\\frac{15}{x-1} = \\frac{10}{y-1} + 1","$$ \\frac{15}{x-1} = \\frac{10}{y-1} + 1 $$","blockMath-mWTPIiVfu",{"__TSPROSE_proseElement":175,"schema":1943,"children":1944,"id":1978},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[1945,1948,1952,1955,1959,1962,1966,1969,1975],{"__TSPROSE_proseElement":175,"schema":1946,"data":1947},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"From ",{"__TSPROSE_proseElement":175,"schema":1949,"data":1950,"storageKey":1890,"id":1951},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1889},"inlinerMath-eyJl4CLbq-1",{"__TSPROSE_proseElement":175,"schema":1953,"data":1954},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", express ",{"__TSPROSE_proseElement":175,"schema":1956,"data":1957,"storageKey":1881,"id":1958},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1880},"inlinerMath-GydTYC7Nc-1",{"__TSPROSE_proseElement":175,"schema":1960,"data":1961},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," through ",{"__TSPROSE_proseElement":175,"schema":1963,"data":1964,"storageKey":443,"id":1965},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":442},"inlinerMath-TLSwLL8iH-8",{"__TSPROSE_proseElement":175,"schema":1967,"data":1968},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},": ",{"__TSPROSE_proseElement":175,"schema":1970,"data":1971,"storageKey":1973,"id":1974},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1972},"y = \\frac{36}{x}","$ y = \\frac{36}{x} $","inlinerMath-k4fFCUP9S",{"__TSPROSE_proseElement":175,"schema":1976,"data":1977},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". 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15x = 10x^2 - 10x + 36x - x^2 - 36 + x \\\\ 540 - 15x = 9x^2 + 27x - 36 \\\\ 0 = 9x^2 + 42x - 576","$$ 540 - 15x = 10x^2 - 10x + 36x - x^2 - 36 + x \\\\ 540 - 15x = 9x^2 + 27x - 36 \\\\ 0 = 9x^2 + 42x - 576 $$","blockMath-DjW4rXQwB",{"__TSPROSE_proseElement":175,"schema":2020,"children":2021,"id":2030},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2022,2024,2028],{"__TSPROSE_proseElement":175,"schema":2023,"data":765},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":2025,"data":2026,"storageKey":660,"id":2027},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"inlinerMath-mqn34wvB0-5",{"__TSPROSE_proseElement":175,"schema":2029,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-mq5DUuH5u",{"__TSPROSE_proseElement":175,"schema":2032,"data":2033,"storageKey":2035,"id":2036},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2034,"freeze":246},"3x^2 + 14x - 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3 \\cdot (-192)}}{3} = \\frac{-7 \\pm \\sqrt{49 + 576}}{3} = \\frac{-7 \\pm \\sqrt{625}}{3} = \\frac{-7 \\pm 25}{3}","$$ x = \\frac{-7 \\pm \\sqrt{7^2 - 3 \\cdot (-192)}}{3} = \\frac{-7 \\pm \\sqrt{49 + 576}}{3} = \\frac{-7 \\pm \\sqrt{625}}{3} = \\frac{-7 \\pm 25}{3} $$","blockMath-KsnpqiRx2",{"__TSPROSE_proseElement":175,"schema":2065,"children":2066,"id":2094},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2067,2070,2076,2078,2084,2087,2091],{"__TSPROSE_proseElement":175,"schema":2068,"data":2069},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"This gives ",{"__TSPROSE_proseElement":175,"schema":2071,"data":2072,"storageKey":2074,"id":2075},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2073},"x = 6","$ x = 6 $","inlinerMath-Aq3ZCMRFS",{"__TSPROSE_proseElement":175,"schema":2077,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":2079,"data":2080,"storageKey":2082,"id":2083},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2081},"x = -\\frac{32}{3}","$ x = -\\frac{32}{3} $","inlinerMath-dJ02tvmsh",{"__TSPROSE_proseElement":175,"schema":2085,"data":2086},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". A negative number of flowers makes no sense, so ",{"__TSPROSE_proseElement":175,"schema":2088,"data":2089,"storageKey":2074,"id":2090},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2073},"inlinerMath-Aq3ZCMRFS-1",{"__TSPROSE_proseElement":175,"schema":2092,"data":2093},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Then:","paragraph-BnysR7Q8L",{"__TSPROSE_proseElement":175,"schema":2096,"data":2097,"storageKey":2099,"id":2100},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2098,"freeze":246},"y = \\frac{36}{x} = \\frac{36}{6} = 6","$$ y = \\frac{36}{x} = \\frac{36}{6} = 6 $$","blockMath-FhIY15Wst",{"__TSPROSE_proseElement":175,"schema":2102,"children":2103,"id":2114},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2104,2107,2111],{"__TSPROSE_proseElement":175,"schema":2105,"data":2106},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"So there are ",{"__TSPROSE_proseElement":175,"schema":2108,"data":2109,"storageKey":619,"id":2110},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":618},"inlinerMath-9rSWhNw25-2",{"__TSPROSE_proseElement":175,"schema":2112,"data":2113},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," flowers along each side. But the spacing differs. Let's compute it:","paragraph-CAhHCSrPc",{"__TSPROSE_proseElement":175,"schema":2116,"data":2117,"storageKey":2119,"id":2120},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2118,"freeze":246},"\\frac{15}{6-1} = \\frac{15}{5} = 3 >>{big} \\frac{10}{6-1} = \\frac{10}{5} = 2","$$ \\frac{15}{6-1} = \\frac{15}{5} = 3 >>{big} \\frac{10}{6-1} = \\frac{10}{5} = 2 $$","blockMath-6KZPmY1de",{"__TSPROSE_proseElement":175,"schema":2122,"children":2123,"id":2146},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2124,2126,2130,2132,2136,2139,2143],{"__TSPROSE_proseElement":175,"schema":2125,"data":1846},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":2127,"data":2128,"storageKey":660,"id":2129},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"inlinerMath-mqn34wvB0-6",{"__TSPROSE_proseElement":175,"schema":2131,"data":1853},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":2133,"data":2134,"storageKey":423,"id":2135},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-7",{"__TSPROSE_proseElement":175,"schema":2137,"data":2138},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meters. The difference really is ",{"__TSPROSE_proseElement":175,"schema":2140,"data":2141,"storageKey":1631,"id":2142},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1630},"inlinerMath-GWJ3eIxvK-3",{"__TSPROSE_proseElement":175,"schema":2144,"data":2145},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meter.","paragraph-d1gwFJiEZ","the-perfectionist-s-garden",{"__TSPROSE_proseElement":175,"schema":2149,"data":2150,"id":2152},{"name":210,"type":245,"linkable":175,"__TSPROSE_schema":175},{"level":263,"title":2151},"Physics","physics",{"__TSPROSE_proseElement":175,"schema":2154,"children":2155,"id":2159},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2156],{"__TSPROSE_proseElement":175,"schema":2157,"data":2158},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"A lot of people hate physics, and that is a shame. Physics gives you one of the biggest piles of examples of situations and problems that reduce to quadratic equations. And you get both kinds here: genuinely real problems and the usual cleverly staged \"puzzles\".","paragraph-USvFKqR3O",{"__TSPROSE_proseElement":175,"schema":2161,"data":2162,"id":2165},{"name":210,"type":245,"linkable":175,"__TSPROSE_schema":175},{"level":2163,"title":2164},2,"Motion with Acceleration","motion-with-acceleration",{"__TSPROSE_proseElement":175,"schema":2167,"children":2168,"id":2189},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2169,2172,2180,2183,2187],{"__TSPROSE_proseElement":175,"schema":2170,"data":2171},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"When forces act on objects, their speed changes: a soccer ball kicked by a foot, a snowball thrown by hand, a bow string launching an arrow, or Earth's gravity speeding up a hammer falling downward. This change in speed can be measured. \"The speed at which speed changes\" sounds weird, but that quantity is called ",{"__TSPROSE_proseElement":175,"schema":2173,"data":2174,"children":2175,"id":2179},{"name":274,"type":255,"linkable":175,"__TSPROSE_schema":175},{"type":276},[2176],{"__TSPROSE_proseElement":175,"schema":2177,"data":2178},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"acceleration","emphasis-8qFXaiBh0",{"__TSPROSE_proseElement":175,"schema":2181,"data":2182},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and is denoted by the letter ",{"__TSPROSE_proseElement":175,"schema":2184,"data":2185,"storageKey":1511,"id":2186},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1510},"inlinerMath-YqDfaaRXA-1",{"__TSPROSE_proseElement":175,"schema":2188,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-1dcoxtWKj",{"__TSPROSE_proseElement":175,"schema":2191,"children":2192,"id":2241},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2193,2196,2202,2205,2211,2214,2220,2223,2229,2232,2238],{"__TSPROSE_proseElement":175,"schema":2194,"data":2195},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The good news is that simple types of accelerated motion are described directly by quadratic equations. No tricky setup is needed at all. The classic example is free fall under Earth's gravity. Our planet gives any object an extra acceleration of ",{"__TSPROSE_proseElement":175,"schema":2197,"data":2198,"storageKey":2200,"id":2201},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2199},"g = 10","$ g = 10 $","inlinerMath-EMd7i6dpk",{"__TSPROSE_proseElement":175,"schema":2203,"data":2204},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meters per second for every second it stays in the air. If an object is dropped from height ",{"__TSPROSE_proseElement":175,"schema":2206,"data":2207,"storageKey":2209,"id":2210},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2208},"h_0","$ h_0 $","inlinerMath-O23hcyX1s",{"__TSPROSE_proseElement":175,"schema":2212,"data":2213},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," with initial speed ",{"__TSPROSE_proseElement":175,"schema":2215,"data":2216,"storageKey":2218,"id":2219},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2217},"v_0","$ v_0 $","inlinerMath-Wuv9SaTZ4",{"__TSPROSE_proseElement":175,"schema":2221,"data":2222},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", then its height ",{"__TSPROSE_proseElement":175,"schema":2224,"data":2225,"storageKey":2227,"id":2228},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2226},"h","$ h $","inlinerMath-1Zm43JKqf",{"__TSPROSE_proseElement":175,"schema":2230,"data":2231},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," at time ",{"__TSPROSE_proseElement":175,"schema":2233,"data":2234,"storageKey":2236,"id":2237},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2235},"t","$ t $","inlinerMath-NdU0V7CGr",{"__TSPROSE_proseElement":175,"schema":2239,"data":2240},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is given by:","paragraph-UmbtQiBPm",{"__TSPROSE_proseElement":175,"schema":2243,"data":2244,"storageKey":2246,"id":2247},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2245,"freeze":246},"h(t) = h_0 + v_0 t - \\frac{g}{2}t^2","$$ h(t) = h_0 + v_0 t - \\frac{g}{2}t^2 $$","blockMath-qcjNUwBzO",{"__TSPROSE_proseElement":175,"schema":2249,"children":2250,"id":2261},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2251,2254,2258],{"__TSPROSE_proseElement":175,"schema":2252,"data":2253},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The sign in front of ",{"__TSPROSE_proseElement":175,"schema":2255,"data":2256,"storageKey":2218,"id":2257},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2217},"inlinerMath-Wuv9SaTZ4-1",{"__TSPROSE_proseElement":175,"schema":2259,"data":2260},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," depends on the direction of the initial velocity. If the object starts downward, you use a minus. If it starts upward, you use a plus.","paragraph-ZJYs9loiX",{"__TSPROSE_proseElement":175,"schema":2263,"data":2264,"children":2268,"id":2680},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":2265},{"title":2266,"level":341,"attributes":2267},"Throwing Balls from the Leaning Tower of Pisa",[],[2269,2301,2307,2313,2330,2354,2619],{"__TSPROSE_proseElement":175,"schema":2270,"children":2271},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[2272,2295],{"__TSPROSE_proseElement":175,"schema":2273,"children":2274,"id":2294},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2275,2278,2284,2287,2291],{"__TSPROSE_proseElement":175,"schema":2276,"data":2277},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Sneaky Galileo climbed the Leaning Tower of Pisa, ",{"__TSPROSE_proseElement":175,"schema":2279,"data":2280,"storageKey":2282,"id":2283},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2281},"60","$ 60 $","inlinerMath-XhWcFzcMl",{"__TSPROSE_proseElement":175,"schema":2285,"data":2286},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meters high, to chuck some balls off it. He does not just drop them -- he throws them at ",{"__TSPROSE_proseElement":175,"schema":2288,"data":2289,"storageKey":796,"id":2290},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":795},"inlinerMath-MAD3UaxRg-1",{"__TSPROSE_proseElement":175,"schema":2292,"data":2293},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," m\u002Fs. He throws the first ball downward and the second upward. After how many seconds does each one hit the ground?","paragraph-QGFXiOZno",{"__TSPROSE_proseElement":175,"schema":2296,"data":2297,"storageKey":2298,"id":2300},{"name":385,"type":245,"linkable":175,"__TSPROSE_schema":175},{"src":2298,"width":2299},"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fpisa-experiment.svg","300px","image-GE3twqDzW",{"__TSPROSE_proseElement":175,"schema":2302,"data":2303},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":2304,"serializedValidator":2305},"Time for first ball to fall",{"__ERUDIT_CHECK":175,"name":396,"data":2306},{"expr":637},{"__TSPROSE_proseElement":175,"schema":2308,"data":2309},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":2310,"serializedValidator":2311},"Time for second ball to fall",{"__ERUDIT_CHECK":175,"name":396,"data":2312},{"expr":609},{"__TSPROSE_proseElement":175,"schema":2314,"children":2315},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[2316],{"__TSPROSE_proseElement":175,"schema":2317,"children":2318,"id":2329},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2319,2322,2326],{"__TSPROSE_proseElement":175,"schema":2320,"data":2321},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"When the ball hits the ground, its final height ",{"__TSPROSE_proseElement":175,"schema":2323,"data":2324,"storageKey":2227,"id":2325},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2226},"inlinerMath-1Zm43JKqf-1",{"__TSPROSE_proseElement":175,"schema":2327,"data":2328},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is zero. Plug that zero and the known values into the height formula.","paragraph-MqRbemvVa",{"__TSPROSE_proseElement":175,"schema":2331,"children":2332},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[2333],{"__TSPROSE_proseElement":175,"schema":2334,"children":2335,"id":2353},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2336,2339,2343,2346,2350],{"__TSPROSE_proseElement":175,"schema":2337,"data":2338},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The first ball hits the ground after ",{"__TSPROSE_proseElement":175,"schema":2340,"data":2341,"storageKey":660,"id":2342},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"inlinerMath-mqn34wvB0-7",{"__TSPROSE_proseElement":175,"schema":2344,"data":2345},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," seconds, and the second after ",{"__TSPROSE_proseElement":175,"schema":2347,"data":2348,"storageKey":610,"id":2349},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":609},"inlinerMath-ZXyaHjumL-3",{"__TSPROSE_proseElement":175,"schema":2351,"data":2352},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," seconds.","paragraph-CGnXOkhz8",{"__TSPROSE_proseElement":175,"schema":2355,"children":2356},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[2357,2489],{"__TSPROSE_proseElement":175,"schema":2358,"data":2360,"children":2361},{"name":2359,"type":245,"linkable":246,"__TSPROSE_schema":175},"problemSection","Thrown downward",[2362,2369,2375,2382,2388,2412,2418,2432,2438,2453,2459],{"__TSPROSE_proseElement":175,"schema":2363,"children":2364,"id":2368},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2365],{"__TSPROSE_proseElement":175,"schema":2366,"data":2367},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The ball is thrown downward, so the initial velocity helps the fall and the height formula becomes:","paragraph-GOh7eS7rG",{"__TSPROSE_proseElement":175,"schema":2370,"data":2371,"storageKey":2373,"id":2374},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2372,"freeze":246},"h(t) = 60 - 5t - \\frac{10}{2}t^2 = 60 - 5t - 5t^2","$$ h(t) = 60 - 5t - \\frac{10}{2}t^2 = 60 - 5t - 5t^2 $$","blockMath-c6bpe6DFw",{"__TSPROSE_proseElement":175,"schema":2376,"children":2377,"id":2381},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2378],{"__TSPROSE_proseElement":175,"schema":2379,"data":2380},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Set the height equal to zero, since that is the moment it hits the ground:","paragraph-OCj7mMhh0",{"__TSPROSE_proseElement":175,"schema":2383,"data":2384,"storageKey":2386,"id":2387},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2385,"freeze":246},"60 - 5t - 5t^2 = 0","$$ 60 - 5t - 5t^2 = 0 $$","blockMath-ciNDIsLli",{"__TSPROSE_proseElement":175,"schema":2389,"children":2390,"id":2411},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2391,2393,2399,2402,2408],{"__TSPROSE_proseElement":175,"schema":2392,"data":765},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":2394,"data":2395,"storageKey":2397,"id":2398},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2396},"-5","$ -5 $","inlinerMath-J3GCr92Iz",{"__TSPROSE_proseElement":175,"schema":2400,"data":2401},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," so the coefficient of ",{"__TSPROSE_proseElement":175,"schema":2403,"data":2404,"storageKey":2406,"id":2407},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2405},"t^2","$ t^2 $","inlinerMath-z3SI7z6Ex",{"__TSPROSE_proseElement":175,"schema":2409,"data":2410},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," becomes positive:","paragraph-sE79WeZZU",{"__TSPROSE_proseElement":175,"schema":2413,"data":2414,"storageKey":2416,"id":2417},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2415,"freeze":246},"t^2 + t - 12 = 0","$$ t^2 + t - 12 = 0 $$","blockMath-rd2qz90GN",{"__TSPROSE_proseElement":175,"schema":2419,"children":2420,"id":2431},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2421,2424,2429],{"__TSPROSE_proseElement":175,"schema":2422,"data":2423},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"We got a quadratic equation. 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It was most likely a ",{"__TSPROSE_proseElement":175,"schema":2646,"data":2647,"children":2648,"id":2652},{"name":274,"type":255,"linkable":175,"__TSPROSE_schema":175},{"type":276},[2649],{"__TSPROSE_proseElement":175,"schema":2650,"data":2651},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"thought experiment","emphasis-oiTboy9Zg",{"__TSPROSE_proseElement":175,"schema":2654,"data":2655},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},": if a heavy object falls faster than a light one, then what happens if you tie them together with a rope? On one hand, the light object should slow the heavy one down. On the other, together they weigh even more than the heavy object alone, so they should fall even faster. Contradiction. 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They hit the lunar surface together, neatly showing that without air resistance, all bodies fall in the same amount of time regardless of mass.","paragraph-71wWzLsYz",{"__TSPROSE_proseElement":175,"schema":2674,"data":2676,"storageKey":2677,"id":2679},{"name":2675,"type":245,"linkable":175,"__TSPROSE_schema":175},"video",{"src":2677,"autoplay":175,"width":2678},"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fhammer-feather.mp4","350px","video-CS3hSMphw","throwing-balls-from-the-leaning-tower-of-pisa",{"__TSPROSE_proseElement":175,"schema":2682,"data":2683,"id":2685},{"name":210,"type":245,"linkable":175,"__TSPROSE_schema":175},{"level":2163,"title":2684},"Electricity","electricity",{"__TSPROSE_proseElement":175,"schema":2687,"children":2688,"id":2719},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2689,2692,2698,2701,2707,2710,2716],{"__TSPROSE_proseElement":175,"schema":2690,"data":2691},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Quadratic equations have also crawled into electric circuits. Circuit elements can be connected in series or in parallel, which lets you flexibly control voltage, current, and resistance. For parallel connections, the total resistance ",{"__TSPROSE_proseElement":175,"schema":2693,"data":2694,"storageKey":2696,"id":2697},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2695},"R_t","$ R_t $","inlinerMath-I2v9y0zZy",{"__TSPROSE_proseElement":175,"schema":2699,"data":2700},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is calculated from the individual resistances ",{"__TSPROSE_proseElement":175,"schema":2702,"data":2703,"storageKey":2705,"id":2706},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2704},"R_1","$ R_1 $","inlinerMath-BvgC7BoEN",{"__TSPROSE_proseElement":175,"schema":2708,"data":2709},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", ",{"__TSPROSE_proseElement":175,"schema":2711,"data":2712,"storageKey":2714,"id":2715},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2713},"R_2","$ R_2 $","inlinerMath-uYvjK2nti",{"__TSPROSE_proseElement":175,"schema":2717,"data":2718},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", and so on by the lovely ugly formula:","paragraph-nzAgPXIqj",{"__TSPROSE_proseElement":175,"schema":2721,"data":2722,"storageKey":2724,"id":2725},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2723,"freeze":246},"\\frac{1}{R_t} = \\frac{1}{R_1} + \\frac{1}{R_2} + \\dots","$$ \\frac{1}{R_t} = \\frac{1}{R_1} + \\frac{1}{R_2} + \\dots $$","blockMath-lxOOF58Mn",{"__TSPROSE_proseElement":175,"schema":2727,"data":2728,"children":2732,"id":2978},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":2729},{"title":2730,"level":341,"attributes":2731},"Fixing a Heater",[],[2733,2763,2773,2783,2807],{"__TSPROSE_proseElement":175,"schema":2734,"children":2735},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[2736,2757],{"__TSPROSE_proseElement":175,"schema":2737,"children":2738,"id":2756},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2739,2742,2746,2749,2753],{"__TSPROSE_proseElement":175,"schema":2740,"data":2741},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"A repairman gets an electric heater made of two heating elements connected in parallel. Everything is sealed in epoxy, so measuring the resistance of each element separately is impossible, but the customer says one element has resistance ",{"__TSPROSE_proseElement":175,"schema":2743,"data":2744,"storageKey":660,"id":2745},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"inlinerMath-mqn34wvB0-10",{"__TSPROSE_proseElement":175,"schema":2747,"data":2748},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," ohms greater than the other. The repairman measures the total resistance of the circuit and gets ",{"__TSPROSE_proseElement":175,"schema":2750,"data":2751,"storageKey":423,"id":2752},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-8",{"__TSPROSE_proseElement":175,"schema":2754,"data":2755},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," ohms. Find the resistance of each heating element.","paragraph-IwnPim3X8",{"__TSPROSE_proseElement":175,"schema":2758,"data":2759,"storageKey":2760,"id":2762},{"name":385,"type":245,"linkable":175,"__TSPROSE_schema":175},{"src":2760,"width":629,"invert":2761},"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fcircuit.svg","dark","image-QSX3w0pRK",{"__TSPROSE_proseElement":175,"schema":2764,"data":2765},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":2766,"serializedValidator":2767},"Element resistances",{"type":1177,"ordered":246,"separator":1178,"answers":2768},[2769,2771],{"__ERUDIT_CHECK":175,"name":396,"data":2770},{"expr":637},{"__ERUDIT_CHECK":175,"name":396,"data":2772},{"expr":618},{"__TSPROSE_proseElement":175,"schema":2774,"children":2775},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[2776],{"__TSPROSE_proseElement":175,"schema":2777,"children":2778,"id":2782},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2779],{"__TSPROSE_proseElement":175,"schema":2780,"data":2781},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Write the larger resistance through the smaller one. Then substitute that into the parallel-resistance formula together with the known total resistance.","paragraph-TFUQhfGc3",{"__TSPROSE_proseElement":175,"schema":2784,"children":2785},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[2786],{"__TSPROSE_proseElement":175,"schema":2787,"children":2788,"id":2806},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2789,2792,2796,2799,2803],{"__TSPROSE_proseElement":175,"schema":2790,"data":2791},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The first element has resistance ",{"__TSPROSE_proseElement":175,"schema":2793,"data":2794,"storageKey":660,"id":2795},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"inlinerMath-mqn34wvB0-11",{"__TSPROSE_proseElement":175,"schema":2797,"data":2798},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," ohms, and the second has resistance ",{"__TSPROSE_proseElement":175,"schema":2800,"data":2801,"storageKey":619,"id":2802},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":618},"inlinerMath-9rSWhNw25-3",{"__TSPROSE_proseElement":175,"schema":2804,"data":2805},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," ohms.","paragraph-i6rDAet3U",{"__TSPROSE_proseElement":175,"schema":2808,"children":2809},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[2810,2824,2830,2844,2850,2869,2875,2915,2921,2953,2959],{"__TSPROSE_proseElement":175,"schema":2811,"children":2812,"id":2823},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2813,2816,2820],{"__TSPROSE_proseElement":175,"schema":2814,"data":2815},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The customer says one element's resistance, call it the second one, is ",{"__TSPROSE_proseElement":175,"schema":2817,"data":2818,"storageKey":660,"id":2819},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":637},"inlinerMath-mqn34wvB0-12",{"__TSPROSE_proseElement":175,"schema":2821,"data":2822},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," ohms greater than the other's. Write that as:","paragraph-3BcmJlmEf",{"__TSPROSE_proseElement":175,"schema":2825,"data":2826,"storageKey":2828,"id":2829},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2827,"freeze":246},"R_2 = R_1 + 3","$$ R_2 = R_1 + 3 $$","blockMath-xP4CTHE6D",{"__TSPROSE_proseElement":175,"schema":2831,"children":2832,"id":2843},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2833,2836,2840],{"__TSPROSE_proseElement":175,"schema":2834,"data":2835},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The total resistance is ",{"__TSPROSE_proseElement":175,"schema":2837,"data":2838,"storageKey":423,"id":2839},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-9",{"__TSPROSE_proseElement":175,"schema":2841,"data":2842},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," ohms. Use the formula for parallel connection to write the equation for the total resistance:","paragraph-r9GIfTS0v",{"__TSPROSE_proseElement":175,"schema":2845,"data":2846,"storageKey":2848,"id":2849},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2847,"freeze":246},"\\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_2}","$$ \\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_2} $$","blockMath-tr8SNBaUb",{"__TSPROSE_proseElement":175,"schema":2851,"children":2852,"id":2868},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2853,2856,2860,2862,2866],{"__TSPROSE_proseElement":175,"schema":2854,"data":2855},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Substitute the expression for ",{"__TSPROSE_proseElement":175,"schema":2857,"data":2858,"storageKey":2714,"id":2859},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2713},"inlinerMath-uYvjK2nti-1",{"__TSPROSE_proseElement":175,"schema":2861,"data":1961},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":2863,"data":2864,"storageKey":2705,"id":2865},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2704},"inlinerMath-BvgC7BoEN-1",{"__TSPROSE_proseElement":175,"schema":2867,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-Jx8Ibxy0B",{"__TSPROSE_proseElement":175,"schema":2870,"data":2871,"storageKey":2873,"id":2874},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2872,"freeze":246},"\\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_1 + 3} \\\\ R_1(R_1 + 3) = 2(R_1 + 3) + 2R_1 \\\\ R_1^2 + 3R_1 = 2R_1 + 6 + 2R_1 \\\\ R_1^2 + 3R_1 = 4R_1 + 6 \\\\ R_1^2 - R_1 - 6 = 0","$$ \\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_1 + 3} \\\\ R_1(R_1 + 3) = 2(R_1 + 3) + 2R_1 \\\\ R_1^2 + 3R_1 = 2R_1 + 6 + 2R_1 \\\\ R_1^2 + 3R_1 = 4R_1 + 6 \\\\ R_1^2 - R_1 - 6 = 0 $$","blockMath-MIlDuivHG",{"__TSPROSE_proseElement":175,"schema":2876,"children":2877,"id":2914},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2878,2881,2886,2888,2892,2894,2900,2902,2906,2908,2912],{"__TSPROSE_proseElement":175,"schema":2879,"data":2880},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"We got a quadratic equation. It can be ",{"__TSPROSE_proseElement":175,"schema":2882,"data":2883,"storageKey":496,"id":2885},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":2884},"factored by hand","dependencyInliner-n70inhAQl",{"__TSPROSE_proseElement":175,"schema":2887,"data":791},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":2889,"data":2890,"storageKey":2555,"id":2891},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2554},"inlinerMath-DLMG5r2qd-1",{"__TSPROSE_proseElement":175,"schema":2893,"data":509},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":2895,"data":2896,"storageKey":2898,"id":2899},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2897},"-6","$ -6 $","inlinerMath-6kVj401tO",{"__TSPROSE_proseElement":175,"schema":2901,"data":518},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":2903,"data":2904,"storageKey":423,"id":2905},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-10",{"__TSPROSE_proseElement":175,"schema":2907,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":2909,"data":2910,"storageKey":820,"id":2911},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":819},"inlinerMath-nfmGVV06J-2",{"__TSPROSE_proseElement":175,"schema":2913,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-XbKWRt6GL",{"__TSPROSE_proseElement":175,"schema":2916,"data":2917,"storageKey":2919,"id":2920},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":2918,"freeze":246},"(R_1 - 3)(R_1 + 2) = 0","$$ (R_1 - 3)(R_1 + 2) = 0 $$","blockMath-NLkCHp4An",{"__TSPROSE_proseElement":175,"schema":2922,"children":2923,"id":2952},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[2924,2927,2933,2936,2942,2945,2949],{"__TSPROSE_proseElement":175,"schema":2925,"data":2926},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"So ",{"__TSPROSE_proseElement":175,"schema":2928,"data":2929,"storageKey":2931,"id":2932},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2930},"R_1 = 3","$ R_1 = 3 $","inlinerMath-Zf0cfVIqT",{"__TSPROSE_proseElement":175,"schema":2934,"data":2935},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," ohms or ",{"__TSPROSE_proseElement":175,"schema":2937,"data":2938,"storageKey":2940,"id":2941},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2939},"R_1 = -2","$ R_1 = -2 $","inlinerMath-pF6Kn5kuv",{"__TSPROSE_proseElement":175,"schema":2943,"data":2944},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," ohms. Negative resistance has no physical meaning, so take ",{"__TSPROSE_proseElement":175,"schema":2946,"data":2947,"storageKey":2931,"id":2948},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2930},"inlinerMath-Zf0cfVIqT-1",{"__TSPROSE_proseElement":175,"schema":2950,"data":2951},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," ohms. 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Even people who hate physics know the simple formula ",{"__TSPROSE_proseElement":175,"schema":2991,"data":2992,"storageKey":2994,"id":2995},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2993},"S = V \\cdot t","$ S = V \\cdot t $","inlinerMath-v58EzLeWz",{"__TSPROSE_proseElement":175,"schema":2997,"data":2998},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", which links distance, speed, and time. It is pretty intuitive that if you sprint to the store for chocolate at ",{"__TSPROSE_proseElement":175,"schema":3000,"data":3001,"storageKey":3003,"id":3004},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3002},"V = 2","$ V = 2 $","inlinerMath-0MVdOOdZg",{"__TSPROSE_proseElement":175,"schema":3006,"data":3007},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," meters per second, then in ",{"__TSPROSE_proseElement":175,"schema":3009,"data":3010,"storageKey":3012,"id":3013},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3011},"t = 10","$ t = 10 $","inlinerMath-tDeedBg6h",{"__TSPROSE_proseElement":175,"schema":3015,"data":3016},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," seconds you will cover ",{"__TSPROSE_proseElement":175,"schema":3018,"data":3019,"storageKey":3021,"id":3022},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3020},"S = V \\cdot t = 2 \\cdot 10 = 20","$ S = V \\cdot t = 2 \\cdot 10 = 20 $","inlinerMath-W4L9uwZhw",{"__TSPROSE_proseElement":175,"schema":3024,"data":942},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-FTyyJxakN",{"__TSPROSE_proseElement":175,"schema":3027,"children":3028,"id":3032},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3029],{"__TSPROSE_proseElement":175,"schema":3030,"data":3031},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"From that basic distance formula, you can derive formulas for speed and time by simple transformations. 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It lets you write an equation connecting two situations: \"round trip time with the current\" = \"trip time in still water\". Express time through distance and speed using ",{"__TSPROSE_proseElement":175,"schema":3436,"data":3437,"storageKey":3439,"id":3440},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3438},"S = Vt","$ S = Vt $","inlinerMath-hns4rdECL",{"__TSPROSE_proseElement":175,"schema":3442,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-WoX3oUVDG",{"__TSPROSE_proseElement":175,"schema":3445,"children":3446},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[3447],{"__TSPROSE_proseElement":175,"schema":3448,"children":3449,"id":3459},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3450,3453,3457],{"__TSPROSE_proseElement":175,"schema":3451,"data":3452},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The kayak's own speed is ",{"__TSPROSE_proseElement":175,"schema":3454,"data":3455,"storageKey":619,"id":3456},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":618},"inlinerMath-9rSWhNw25-7",{"__TSPROSE_proseElement":175,"schema":3458,"data":3308},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-y6f1Wehds",{"__TSPROSE_proseElement":175,"schema":3461,"children":3462},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[3463,3480,3486,3493,3498,3511,3517,3524,3530,3561,3567,3581,3587,3600,3606,3619,3625],{"__TSPROSE_proseElement":175,"schema":3464,"children":3465,"id":3479},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3466,3469,3476],{"__TSPROSE_proseElement":175,"schema":3467,"data":3468},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The important thing is the phrase ",{"__TSPROSE_proseElement":175,"schema":3470,"data":3471,"children":3472,"id":3475},{"name":274,"type":255,"linkable":175,"__TSPROSE_schema":175},{"type":276},[3473],{"__TSPROSE_proseElement":175,"schema":3474,"data":3430},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"emphasis-fPku1DuoL-1",{"__TSPROSE_proseElement":175,"schema":3477,"data":3478},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". We do not know the actual time, but we do know it is equal in both situations. That lets us equate the \"upstream plus downstream trip with the current\" to the \"trip in still water\":","paragraph-BjkYZLq8T",{"__TSPROSE_proseElement":175,"schema":3481,"data":3482,"storageKey":3484,"id":3485},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3483,"freeze":246},"\\boxed{t_{\\text{upstream}} + t_{\\text{downstream}} = t_{\\text{still}}}","$$ \\boxed{t_{\\text{upstream}} + t_{\\text{downstream}} = t_{\\text{still}}} $$","blockMath-ucJF0EZlo",{"__TSPROSE_proseElement":175,"schema":3487,"children":3488,"id":3492},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3489],{"__TSPROSE_proseElement":175,"schema":3490,"data":3491},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"But the problem does not directly give any times. What it does give are distances and speeds. So we need to express time through those. Start from the standard formula:","paragraph-FVIs9L82T",{"__TSPROSE_proseElement":175,"schema":3494,"data":3495,"storageKey":3496,"id":3497},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3438,"freeze":246},"$$ S = Vt $$","blockMath-d9Kkt7CCh",{"__TSPROSE_proseElement":175,"schema":3499,"children":3500,"id":3510},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3501,3503,3507],{"__TSPROSE_proseElement":175,"schema":3502,"data":765},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":3504,"data":3505,"storageKey":3142,"id":3506},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3141},"inlinerMath-zPgGLZYmJ-1",{"__TSPROSE_proseElement":175,"schema":3508,"data":3509},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," to isolate time:","paragraph-iEHfIi5Ta",{"__TSPROSE_proseElement":175,"schema":3512,"data":3513,"storageKey":3515,"id":3516},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3514,"freeze":246},"\\frac{S}{V} = \\frac{\\cancel{V}t}{\\cancel{V}} >>{big} \\boxed{t = \\frac{S}{V}}","$$ \\frac{S}{V} = \\frac{\\cancel{V}t}{\\cancel{V}} >>{big} \\boxed{t = \\frac{S}{V}} $$","blockMath-TrEit9oi5",{"__TSPROSE_proseElement":175,"schema":3518,"children":3519,"id":3523},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3520],{"__TSPROSE_proseElement":175,"schema":3521,"data":3522},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Substitute those time expressions into the equation:","paragraph-MS7js23Z0",{"__TSPROSE_proseElement":175,"schema":3525,"data":3526,"storageKey":3528,"id":3529},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3527,"freeze":246},"\\frac{S_{\\text{upstream}}}{V_{\\text{upstream}}} + \\frac{S_{\\text{downstream}}}{V_{\\text{downstream}}} = \\frac{S_{\\text{still}}}{V_{\\text{still}}}","$$ \\frac{S_{\\text{upstream}}}{V_{\\text{upstream}}} + \\frac{S_{\\text{downstream}}}{V_{\\text{downstream}}} = \\frac{S_{\\text{still}}}{V_{\\text{still}}} $$","blockMath-gMRpMc5Vk",{"__TSPROSE_proseElement":175,"schema":3531,"children":3532,"id":3560},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3533,3536,3542,3545,3549,3552,3558],{"__TSPROSE_proseElement":175,"schema":3534,"data":3535},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The distances are known. Upstream speed equals the kayak's own speed ",{"__TSPROSE_proseElement":175,"schema":3537,"data":3538,"storageKey":3540,"id":3541},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3539},"V_б","$ V_б $","inlinerMath-jbyOb7IEx",{"__TSPROSE_proseElement":175,"schema":3543,"data":3544},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," minus the current speed ",{"__TSPROSE_proseElement":175,"schema":3546,"data":3547,"storageKey":1631,"id":3548},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1630},"inlinerMath-GWJ3eIxvK-5",{"__TSPROSE_proseElement":175,"schema":3550,"data":3551},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Downstream speed equals ",{"__TSPROSE_proseElement":175,"schema":3553,"data":3554,"storageKey":3556,"id":3557},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3555},"V_б + 1","$ V_б + 1 $","inlinerMath-RPMs33uiJ",{"__TSPROSE_proseElement":175,"schema":3559,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-kQXHaWDnY",{"__TSPROSE_proseElement":175,"schema":3562,"data":3563,"storageKey":3565,"id":3566},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3564,"freeze":246},"\\frac{15}{V_б - 1} + \\frac{14}{V_б + 1} = \\frac{30}{V_б}","$$ \\frac{15}{V_б - 1} + \\frac{14}{V_б + 1} = \\frac{30}{V_б} $$","blockMath-DBKsVLAAt",{"__TSPROSE_proseElement":175,"schema":3568,"children":3569,"id":3580},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3570,3573,3577],{"__TSPROSE_proseElement":175,"schema":3571,"data":3572},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"We got an equation with one unknown, which is exactly the kayak's own speed ",{"__TSPROSE_proseElement":175,"schema":3574,"data":3575,"storageKey":3540,"id":3576},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3539},"inlinerMath-jbyOb7IEx-1",{"__TSPROSE_proseElement":175,"schema":3578,"data":3579},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Simplify it:","paragraph-caTtGuHf8",{"__TSPROSE_proseElement":175,"schema":3582,"data":3583,"storageKey":3585,"id":3586},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3584,"freeze":246},"15(V_б + 1)V_б + 14(V_б - 1)V_б = 30(V_б + 1)(V_б - 1) \\\\ 15(V_б^2 + V_б) + 14(V_б^2 - V_б) = 30(V_б^2 - 1) \\\\ 15V_б^2 + 15V_б + 14V_б^2 - 14V_б = 30V_б^2 - 30 \\\\ 29V_б^2 + V_б = 30V_б^2 - 30 \\\\ 0 = 30V_б^2 - 29V_б^2 - V_б - 30 \\\\ 0 = V_б^2 - V_б - 30","$$ 15(V_б + 1)V_б + 14(V_б - 1)V_б = 30(V_б + 1)(V_б - 1) \\\\ 15(V_б^2 + V_б) + 14(V_б^2 - V_б) = 30(V_б^2 - 1) \\\\ 15V_б^2 + 15V_б + 14V_б^2 - 14V_б = 30V_б^2 - 30 \\\\ 29V_б^2 + V_б = 30V_б^2 - 30 \\\\ 0 = 30V_б^2 - 29V_б^2 - V_б - 30 \\\\ 0 = V_б^2 - V_б - 30 $$","blockMath-W8LPakxhU",{"__TSPROSE_proseElement":175,"schema":3588,"children":3589,"id":3599},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3590,3593,3597],{"__TSPROSE_proseElement":175,"schema":3591,"data":3592},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Now find the ",{"__TSPROSE_proseElement":175,"schema":3594,"data":3595,"storageKey":2427,"id":3596},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":193},"dependencyInliner-emFvGyx3M-2",{"__TSPROSE_proseElement":175,"schema":3598,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-AirhIkDv2",{"__TSPROSE_proseElement":175,"schema":3601,"data":3602,"storageKey":3604,"id":3605},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3603,"freeze":246},"D = (-1)^2 - 4 \\cdot 1 \\cdot (-30) = 1 + 120 = 121","$$ D = (-1)^2 - 4 \\cdot 1 \\cdot (-30) = 1 + 120 = 121 $$","blockMath-1nIYdp6c7",{"__TSPROSE_proseElement":175,"schema":3607,"children":3608,"id":3618},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3609,3612,3616],{"__TSPROSE_proseElement":175,"schema":3610,"data":3611},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"And the roots with the ",{"__TSPROSE_proseElement":175,"schema":3613,"data":3614,"storageKey":2448,"id":3615},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":2447},"dependencyInliner-5jM01BIgz-1",{"__TSPROSE_proseElement":175,"schema":3617,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-LXe6N5W2z",{"__TSPROSE_proseElement":175,"schema":3620,"data":3621,"storageKey":3623,"id":3624},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3622,"freeze":246},"V_б = \\frac{-(-1) \\pm \\sqrt{121}}{2 \\cdot 1} = \\frac{1 \\pm 11}{2}","$$ V_б = \\frac{-(-1) \\pm \\sqrt{121}}{2 \\cdot 1} = \\frac{1 \\pm 11}{2} $$","blockMath-tdym9Z4a7",{"__TSPROSE_proseElement":175,"schema":3626,"children":3627,"id":3653},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3628,3630,3636,3638,3644,3647,3651],{"__TSPROSE_proseElement":175,"schema":3629,"data":2593},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":3631,"data":3632,"storageKey":3634,"id":3635},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3633},"V_б = 6","$ V_б = 6 $","inlinerMath-QcD1sGBlW",{"__TSPROSE_proseElement":175,"schema":3637,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":3639,"data":3640,"storageKey":3642,"id":3643},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3641},"V_б = -5","$ V_б = -5 $","inlinerMath-4olNkShnE",{"__TSPROSE_proseElement":175,"schema":3645,"data":3646},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Negative speed is physically meaningless, so the kayak's own speed is ",{"__TSPROSE_proseElement":175,"schema":3648,"data":3649,"storageKey":619,"id":3650},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":618},"inlinerMath-9rSWhNw25-8",{"__TSPROSE_proseElement":175,"schema":3652,"data":3308},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-KOhdu4zzy",{"__TSPROSE_proseElement":175,"schema":3655,"data":3656,"children":3657},{"name":3367,"type":245,"linkable":246,"__TSPROSE_schema":175},{},[3658,3691,3697,3713,3928],{"__TSPROSE_proseElement":175,"schema":3659,"children":3660},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[3661],{"__TSPROSE_proseElement":175,"schema":3662,"children":3663,"id":3690},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3664,3667,3673,3676,3680,3683,3687],{"__TSPROSE_proseElement":175,"schema":3665,"data":3666},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"A train needs to travel ",{"__TSPROSE_proseElement":175,"schema":3668,"data":3669,"storageKey":3671,"id":3672},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3670},"200","$ 200 $","inlinerMath-TpmsrDaej",{"__TSPROSE_proseElement":175,"schema":3674,"data":3675},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," kilometers at constant speed. If it went ",{"__TSPROSE_proseElement":175,"schema":3677,"data":3678,"storageKey":1085,"id":3679},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1084},"inlinerMath-8Y3Z60Vbg-3",{"__TSPROSE_proseElement":175,"schema":3681,"data":3682},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," km\u002Fh faster, the trip would take ",{"__TSPROSE_proseElement":175,"schema":3684,"data":3685,"storageKey":1631,"id":3686},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1630},"inlinerMath-GWJ3eIxvK-6",{"__TSPROSE_proseElement":175,"schema":3688,"data":3689},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hour less. Find the train's speed.","paragraph-Woz7eKnBk",{"__TSPROSE_proseElement":175,"schema":3692,"data":3693},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":3694,"serializedValidator":3695},"Train speed",{"__ERUDIT_CHECK":175,"name":396,"data":3696},{"expr":358},{"__TSPROSE_proseElement":175,"schema":3698,"children":3699},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[3700],{"__TSPROSE_proseElement":175,"schema":3701,"children":3702,"id":3712},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3703,3706,3710],{"__TSPROSE_proseElement":175,"schema":3704,"data":3705},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The train's speed is ",{"__TSPROSE_proseElement":175,"schema":3707,"data":3708,"storageKey":359,"id":3709},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":358},"inlinerMath-9HaCbxMgV-1",{"__TSPROSE_proseElement":175,"schema":3711,"data":3308},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-RJmQpPKaj",{"__TSPROSE_proseElement":175,"schema":3714,"children":3715},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[3716,3765,3772,3777,3807,3813,3828,3834,3876,3882,3910,3916],{"__TSPROSE_proseElement":175,"schema":3717,"children":3718,"id":3764},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3719,3722,3726,3729,3735,3738,3744,3747,3753,3756,3762],{"__TSPROSE_proseElement":175,"schema":3720,"data":3721},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"There are two situations. In the first, the train goes ",{"__TSPROSE_proseElement":175,"schema":3723,"data":3724,"storageKey":3671,"id":3725},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3670},"inlinerMath-TpmsrDaej-1",{"__TSPROSE_proseElement":175,"schema":3727,"data":3728},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," kilometers at speed ",{"__TSPROSE_proseElement":175,"schema":3730,"data":3731,"storageKey":3733,"id":3734},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3732},"V_1","$ V_1 $","inlinerMath-5rWU3hEgh",{"__TSPROSE_proseElement":175,"schema":3736,"data":3737},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and takes ",{"__TSPROSE_proseElement":175,"schema":3739,"data":3740,"storageKey":3742,"id":3743},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3741},"t_1","$ t_1 $","inlinerMath-tbk2hBmyF",{"__TSPROSE_proseElement":175,"schema":3745,"data":3746},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hours. In the second, the distance is the same, but the speed is ",{"__TSPROSE_proseElement":175,"schema":3748,"data":3749,"storageKey":3751,"id":3752},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3750},"V_2 = V_1 + 10","$ V_2 = V_1 + 10 $","inlinerMath-RpOSLzEsV",{"__TSPROSE_proseElement":175,"schema":3754,"data":3755},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", and the travel time is one hour shorter, that is, ",{"__TSPROSE_proseElement":175,"schema":3757,"data":3758,"storageKey":3760,"id":3761},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3759},"t_2 = t_1 - 1","$ t_2 = t_1 - 1 $","inlinerMath-jZooVJcKL",{"__TSPROSE_proseElement":175,"schema":3763,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-Iktb4i57C",{"__TSPROSE_proseElement":175,"schema":3766,"children":3767,"id":3771},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3768],{"__TSPROSE_proseElement":175,"schema":3769,"data":3770},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"For variety, this time we build the main equation using speed instead of time:","paragraph-cqFv0ER12",{"__TSPROSE_proseElement":175,"schema":3773,"data":3774,"storageKey":3775,"id":3776},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3750,"freeze":246},"$$ V_2 = V_1 + 10 $$","blockMath-Zb2G9fqmf",{"__TSPROSE_proseElement":175,"schema":3778,"children":3779,"id":3806},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3780,3783,3787,3790,3794,3797,3803],{"__TSPROSE_proseElement":175,"schema":3781,"data":3782},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Express speed through distance and time. From ",{"__TSPROSE_proseElement":175,"schema":3784,"data":3785,"storageKey":3439,"id":3786},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3438},"inlinerMath-hns4rdECL-1",{"__TSPROSE_proseElement":175,"schema":3788,"data":3789},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", divide both sides by ",{"__TSPROSE_proseElement":175,"schema":3791,"data":3792,"storageKey":2236,"id":3793},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2235},"inlinerMath-NdU0V7CGr-1",{"__TSPROSE_proseElement":175,"schema":3795,"data":3796},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". That gives ",{"__TSPROSE_proseElement":175,"schema":3798,"data":3799,"storageKey":3801,"id":3802},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3800},"V = \\frac{S}{t}","$ V = \\frac{S}{t} $","inlinerMath-hNGI5f6cH",{"__TSPROSE_proseElement":175,"schema":3804,"data":3805},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Substitute that into the equation above:","paragraph-7b0c13iiJ",{"__TSPROSE_proseElement":175,"schema":3808,"data":3809,"storageKey":3811,"id":3812},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3810,"freeze":246},"\\frac{S}{t_2} = \\frac{S}{t_1} + 10 \\\\ \\frac{200}{t_1 - 1} = \\frac{200}{t_1} + 10","$$ \\frac{S}{t_2} = \\frac{S}{t_1} + 10 \\\\ \\frac{200}{t_1 - 1} = \\frac{200}{t_1} + 10 $$","blockMath-CyLJgojzK",{"__TSPROSE_proseElement":175,"schema":3814,"children":3815,"id":3827},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3816,3818,3824],{"__TSPROSE_proseElement":175,"schema":3817,"data":1990},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":3819,"data":3820,"storageKey":3822,"id":3823},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3821},"t_1(t_1 - 1)","$ t_1(t_1 - 1) $","inlinerMath-agJ7sQvza",{"__TSPROSE_proseElement":175,"schema":3825,"data":3826},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," to remove fractions:","paragraph-IgvN6gLxk",{"__TSPROSE_proseElement":175,"schema":3829,"data":3830,"storageKey":3832,"id":3833},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3831,"freeze":246},"200t_1 = 200(t_1 - 1) + 10t_1(t_1 - 1) \\\\ 200t_1 = 200t_1 - 200 + 10t_1^2 - 10t_1 \\\\ 0 = 10t_1^2 - 10t_1 - 200 \\\\ 0 = t_1^2 - t_1 - 20","$$ 200t_1 = 200(t_1 - 1) + 10t_1(t_1 - 1) \\\\ 200t_1 = 200t_1 - 200 + 10t_1^2 - 10t_1 \\\\ 0 = 10t_1^2 - 10t_1 - 200 \\\\ 0 = t_1^2 - t_1 - 20 $$","blockMath-8OvpCGLDs",{"__TSPROSE_proseElement":175,"schema":3835,"children":3836,"id":3875},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3837,3840,3845,3848,3852,3855,3861,3863,3867,3869,3873],{"__TSPROSE_proseElement":175,"schema":3838,"data":3839},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"You can ",{"__TSPROSE_proseElement":175,"schema":3841,"data":3842,"storageKey":496,"id":3844},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":3843},"factor this by hand","dependencyInliner-wCRZqkthW",{"__TSPROSE_proseElement":175,"schema":3846,"data":3847},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". We need one positive and one negative number whose sum is ",{"__TSPROSE_proseElement":175,"schema":3849,"data":3850,"storageKey":2555,"id":3851},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2554},"inlinerMath-DLMG5r2qd-2",{"__TSPROSE_proseElement":175,"schema":3853,"data":3854},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and product is ",{"__TSPROSE_proseElement":175,"schema":3856,"data":3857,"storageKey":3859,"id":3860},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3858},"-20","$ -20 $","inlinerMath-eqj9W41JY",{"__TSPROSE_proseElement":175,"schema":3862,"data":518},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":3864,"data":3865,"storageKey":610,"id":3866},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":609},"inlinerMath-ZXyaHjumL-5",{"__TSPROSE_proseElement":175,"schema":3868,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":3870,"data":3871,"storageKey":2397,"id":3872},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2396},"inlinerMath-J3GCr92Iz-2",{"__TSPROSE_proseElement":175,"schema":3874,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-jBWwrluyn",{"__TSPROSE_proseElement":175,"schema":3877,"data":3878,"storageKey":3880,"id":3881},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3879,"freeze":246},"(t_1 + 4)(t_1 - 5) = 0","$$ (t_1 + 4)(t_1 - 5) = 0 $$","blockMath-ZXemnTIcn",{"__TSPROSE_proseElement":175,"schema":3883,"children":3884,"id":3909},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3885,3888,3892,3895,3899,3902,3906],{"__TSPROSE_proseElement":175,"schema":3886,"data":3887},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"So the original trip time could be ",{"__TSPROSE_proseElement":175,"schema":3889,"data":3890,"storageKey":2577,"id":3891},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2576},"inlinerMath-aM1jeM3he-1",{"__TSPROSE_proseElement":175,"schema":3893,"data":3894},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hours or ",{"__TSPROSE_proseElement":175,"schema":3896,"data":3897,"storageKey":796,"id":3898},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":795},"inlinerMath-MAD3UaxRg-2",{"__TSPROSE_proseElement":175,"schema":3900,"data":3901},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hours. Negative time is nonsense, so the trip lasted ",{"__TSPROSE_proseElement":175,"schema":3903,"data":3904,"storageKey":796,"id":3905},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":795},"inlinerMath-MAD3UaxRg-3",{"__TSPROSE_proseElement":175,"schema":3907,"data":3908},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hours. Then the speed is:","paragraph-bfkUJjYyl",{"__TSPROSE_proseElement":175,"schema":3911,"data":3912,"storageKey":3914,"id":3915},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":3913,"freeze":246},"S = Vt >>{big} \\frac{S}{t} = V >>{big} V = \\frac{200}{5} = 40","$$ S = Vt >>{big} \\frac{S}{t} = V >>{big} V = \\frac{200}{5} = 40 $$","blockMath-kQHbgur5Y",{"__TSPROSE_proseElement":175,"schema":3917,"children":3918,"id":3927},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3919,3921,3925],{"__TSPROSE_proseElement":175,"schema":3920,"data":3705},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":3922,"data":3923,"storageKey":359,"id":3924},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":358},"inlinerMath-9HaCbxMgV-2",{"__TSPROSE_proseElement":175,"schema":3926,"data":3308},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-RJmQpPKaj-1",{"__TSPROSE_proseElement":175,"schema":3929,"children":3930},{"name":2621,"type":245,"linkable":246,"__TSPROSE_schema":175},[3931],{"__TSPROSE_proseElement":175,"schema":3932,"children":3933,"id":3953},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3934,3937,3941,3944,3950],{"__TSPROSE_proseElement":175,"schema":3935,"data":3936},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"As you can see, problems like this can be solved in different ways depending on what quantity you lean on. In this case it would obviously be more convenient to build the equation from ",{"__TSPROSE_proseElement":175,"schema":3938,"data":3939,"storageKey":3760,"id":3940},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3759},"inlinerMath-jZooVJcKL-1",{"__TSPROSE_proseElement":175,"schema":3942,"data":3943},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", because then we would substitute expressions like ",{"__TSPROSE_proseElement":175,"schema":3945,"data":3946,"storageKey":3948,"id":3949},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3947},"\\frac{S}{V}","$ \\frac{S}{V} $","inlinerMath-aC7fWCpBM",{"__TSPROSE_proseElement":175,"schema":3951,"data":3952},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and look for the speed directly. But it is not a disaster to build it through speed, find the time, and then get the speed from that.","paragraph-8EJLYgzzU",{"__TSPROSE_proseElement":175,"schema":3955,"data":3956,"children":3957},{"name":3367,"type":245,"linkable":246,"__TSPROSE_schema":175},{},[3958,3991,3997,4013],{"__TSPROSE_proseElement":175,"schema":3959,"children":3960},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[3961],{"__TSPROSE_proseElement":175,"schema":3962,"children":3963,"id":3990},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[3964,3967,3971,3974,3978,3981,3987],{"__TSPROSE_proseElement":175,"schema":3965,"data":3966},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"A river ferry can reach ",{"__TSPROSE_proseElement":175,"schema":3968,"data":3969,"storageKey":1085,"id":3970},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1084},"inlinerMath-8Y3Z60Vbg-4",{"__TSPROSE_proseElement":175,"schema":3972,"data":3973},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," km\u002Fh in still water. Find the current speed if going ",{"__TSPROSE_proseElement":175,"schema":3975,"data":3976,"storageKey":1794,"id":3977},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1793},"inlinerMath-84Rv0uj32-1",{"__TSPROSE_proseElement":175,"schema":3979,"data":3980},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," kilometers upstream takes ",{"__TSPROSE_proseElement":175,"schema":3982,"data":3983,"storageKey":3985,"id":3986},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3984},"1.5","$ 1.5 $","inlinerMath-nHi4DfuOB",{"__TSPROSE_proseElement":175,"schema":3988,"data":3989},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hours longer than coming back.","paragraph-TBoeHRGwD",{"__TSPROSE_proseElement":175,"schema":3992,"data":3993},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":3994,"serializedValidator":3995},"Current speed",{"__ERUDIT_CHECK":175,"name":396,"data":3996},{"expr":398},{"__TSPROSE_proseElement":175,"schema":3998,"children":3999},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[4000],{"__TSPROSE_proseElement":175,"schema":4001,"children":4002,"id":4012},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4003,4006,4010],{"__TSPROSE_proseElement":175,"schema":4004,"data":4005},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The current speed is ",{"__TSPROSE_proseElement":175,"schema":4007,"data":4008,"storageKey":423,"id":4009},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-14",{"__TSPROSE_proseElement":175,"schema":4011,"data":3308},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-uqHVmGVb2",{"__TSPROSE_proseElement":175,"schema":4014,"children":4015},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[4016,4050,4056,4069,4075,4089,4095],{"__TSPROSE_proseElement":175,"schema":4017,"children":4018,"id":4049},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4019,4022,4028,4031,4037,4040,4046],{"__TSPROSE_proseElement":175,"schema":4020,"data":4021},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Let the upstream travel time be ",{"__TSPROSE_proseElement":175,"schema":4023,"data":4024,"storageKey":4026,"id":4027},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4025},"t_{\\uarr}","$ t_{\\uarr} $","inlinerMath-ygtLSQvlc",{"__TSPROSE_proseElement":175,"schema":4029,"data":4030},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", the downstream time be ",{"__TSPROSE_proseElement":175,"schema":4032,"data":4033,"storageKey":4035,"id":4036},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4034},"t_{\\darr}","$ t_{\\darr} $","inlinerMath-Tn6sR3SgT",{"__TSPROSE_proseElement":175,"schema":4038,"data":4039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", and the current speed be ",{"__TSPROSE_proseElement":175,"schema":4041,"data":4042,"storageKey":4044,"id":4045},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4043},"V_\\approx","$ V_\\approx $","inlinerMath-8AflEoznk",{"__TSPROSE_proseElement":175,"schema":4047,"data":4048},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Again, we use the time relation given in the problem:","paragraph-tRy4tdiQZ",{"__TSPROSE_proseElement":175,"schema":4051,"data":4052,"storageKey":4054,"id":4055},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":4053,"freeze":246},"t_{\\darr} = t_{\\uarr} - \\underbrace{1.5}_{\\small\\frac{3}{2}}","$$ t_{\\darr} = t_{\\uarr} - \\underbrace{1.5}_{\\small\\frac{3}{2}} $$","blockMath-CKfbKOrho",{"__TSPROSE_proseElement":175,"schema":4057,"children":4058,"id":4068},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4059,4062,4066],{"__TSPROSE_proseElement":175,"schema":4060,"data":4061},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Since the actual times are unknown, substitute expressions for time through distance and speed,",{"__TSPROSE_proseElement":175,"schema":4063,"data":4064,"storageKey":3948,"id":4065},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3947},"inlinerMath-aC7fWCpBM-1",{"__TSPROSE_proseElement":175,"schema":4067,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-R9IcRdiWv",{"__TSPROSE_proseElement":175,"schema":4070,"data":4071,"storageKey":4073,"id":4074},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":4072,"freeze":246},"\\frac{36}{10 + V_\\approx} = \\frac{36}{10 - V_\\approx} - \\frac{3}{2} \\\\ 36(10 - V_\\approx) = 36(10 + V_\\approx) - \\frac{3}{2}(10 + V_\\approx)(10 - V_\\approx) \\\\ \\cancel{360} - 36V_\\approx = \\cancel{360} + 36V_\\approx - \\frac{3}{2}(100 - V_\\approx^2) \\\\ -72V_\\approx = - \\frac{3}{2}(100 - V_\\approx^2) \\\\ -24V_\\approx = -\\frac{1}{2}(100 - V_\\approx^2) \\\\ 48V_\\approx = 100 - V_\\approx^2 \\\\ V_\\approx^2 + 48V_\\approx - 100 = 0","$$ \\frac{36}{10 + V_\\approx} = \\frac{36}{10 - V_\\approx} - \\frac{3}{2} \\\\ 36(10 - V_\\approx) = 36(10 + V_\\approx) - \\frac{3}{2}(10 + V_\\approx)(10 - V_\\approx) \\\\ \\cancel{360} - 36V_\\approx = \\cancel{360} + 36V_\\approx - \\frac{3}{2}(100 - V_\\approx^2) \\\\ -72V_\\approx = - \\frac{3}{2}(100 - V_\\approx^2) \\\\ -24V_\\approx = -\\frac{1}{2}(100 - V_\\approx^2) \\\\ 48V_\\approx = 100 - V_\\approx^2 \\\\ V_\\approx^2 + 48V_\\approx - 100 = 0 $$","blockMath-w4ljqgh7K",{"__TSPROSE_proseElement":175,"schema":4076,"children":4077,"id":4088},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4078,4081,4086],{"__TSPROSE_proseElement":175,"schema":4079,"data":4080},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Yet another simple quadratic equation that can be solved by ",{"__TSPROSE_proseElement":175,"schema":4082,"data":4083,"storageKey":496,"id":4085},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":4084},"factoring by hand","dependencyInliner-GSXziuB7c",{"__TSPROSE_proseElement":175,"schema":4087,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-7bOuNd4gT",{"__TSPROSE_proseElement":175,"schema":4090,"data":4091,"storageKey":4093,"id":4094},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":4092,"freeze":246},"(V_\\approx + 50)(V_\\approx - 2) = 0","$$ (V_\\approx + 50)(V_\\approx - 2) = 0 $$","blockMath-7XAOThioJ",{"__TSPROSE_proseElement":175,"schema":4096,"children":4097,"id":4116},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4098,4101,4107,4110,4114],{"__TSPROSE_proseElement":175,"schema":4099,"data":4100},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"A current speed of ",{"__TSPROSE_proseElement":175,"schema":4102,"data":4103,"storageKey":4105,"id":4106},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4104},"-50","$ -50 $","inlinerMath-fVCmx5sKN",{"__TSPROSE_proseElement":175,"schema":4108,"data":4109},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," km\u002Fh is nonsense, so the actual current speed is ",{"__TSPROSE_proseElement":175,"schema":4111,"data":4112,"storageKey":423,"id":4113},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-15",{"__TSPROSE_proseElement":175,"schema":4115,"data":3308},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-q0JRIazxm","hunting-for-speed",{"__TSPROSE_proseElement":175,"schema":4119,"data":4120,"children":4124,"id":4406},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":4121},{"title":4122,"level":1141,"attributes":4123},"Long Hike",[],[4125,4156,4162,4172,4189],{"__TSPROSE_proseElement":175,"schema":4126,"children":4127},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[4128],{"__TSPROSE_proseElement":175,"schema":4129,"children":4130,"id":4155},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4131,4134,4138,4141,4145,4148,4152],{"__TSPROSE_proseElement":175,"schema":4132,"data":4133},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"A tourist went on a hike. The route is a decent one -- ",{"__TSPROSE_proseElement":175,"schema":4135,"data":4136,"storageKey":359,"id":4137},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":358},"inlinerMath-9HaCbxMgV-3",{"__TSPROSE_proseElement":175,"schema":4139,"data":4140},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," kilometers long. 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Negative drone speed belongs in some parallel universe, not here, so the real speed is ",{"__TSPROSE_proseElement":175,"schema":4674,"data":4675,"storageKey":4677,"id":4678},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4676},"9","$ 9 $","inlinerMath-Ss7zKstt6",{"__TSPROSE_proseElement":175,"schema":4680,"data":4681},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," km\u002Fh. 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Speed ",{"__TSPROSE_proseElement":175,"schema":4752,"data":4753,"storageKey":3142,"id":4754},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3141},"inlinerMath-zPgGLZYmJ-3",{"__TSPROSE_proseElement":175,"schema":4756,"data":4757},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," becomes \"productivity\", \"output\", or just a work rate ",{"__TSPROSE_proseElement":175,"schema":4759,"data":4760,"storageKey":1277,"id":4761},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1276},"inlinerMath-NNMK5da4D-1",{"__TSPROSE_proseElement":175,"schema":4763,"data":4764},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},": how many square meters get painted per hour, how many liters flow into a tank per minute, how many parts get made per day, and so on.","paragraph-rDGUXVmEN",{"__TSPROSE_proseElement":175,"schema":4767,"children":4768,"id":4806},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4769,4772,4778,4781,4785,4788,4794,4797,4803],{"__TSPROSE_proseElement":175,"schema":4770,"data":4771},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The formulas are exactly the same, only the letters change. For example, if a worker makes ",{"__TSPROSE_proseElement":175,"schema":4773,"data":4774,"storageKey":4776,"id":4777},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4775},"P = 8","$ P = 8 $","inlinerMath-QSvrYc3bu",{"__TSPROSE_proseElement":175,"schema":4779,"data":4780},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," parts per hour, then in ",{"__TSPROSE_proseElement":175,"schema":4782,"data":4783,"storageKey":3012,"id":4784},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3011},"inlinerMath-tDeedBg6h-1",{"__TSPROSE_proseElement":175,"schema":4786,"data":4787},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hours they make ",{"__TSPROSE_proseElement":175,"schema":4789,"data":4790,"storageKey":4792,"id":4793},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4791},"8 \\cdot 10 = 80 = W","$ 8 \\cdot 10 = 80 = W $","inlinerMath-phWErnaWh",{"__TSPROSE_proseElement":175,"schema":4795,"data":4796},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," parts. 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How many hours would each worker need on their own?","paragraph-fvEk7xW2N",{"__TSPROSE_proseElement":175,"schema":4845,"data":4846},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":4847,"serializedValidator":4848},"Individual painting times",{"type":1177,"ordered":246,"separator":1178,"answers":4849},[4850,4852],{"__ERUDIT_CHECK":175,"name":396,"data":4851},{"expr":1084},{"__ERUDIT_CHECK":175,"name":396,"data":4853},{"expr":1784},{"__TSPROSE_proseElement":175,"schema":4855,"children":4856},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[4857],{"__TSPROSE_proseElement":175,"schema":4858,"children":4859,"id":4870},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4860,4863,4867],{"__TSPROSE_proseElement":175,"schema":4861,"data":4862},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Start from the work formula ",{"__TSPROSE_proseElement":175,"schema":4864,"data":4865,"storageKey":4801,"id":4866},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4800},"inlinerMath-xPOEN3cy3-1",{"__TSPROSE_proseElement":175,"schema":4868,"data":4869},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," for the situation where both workers are painting together. 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They can paint the fence in ",{"__TSPROSE_proseElement":175,"schema":4904,"data":4905,"storageKey":619,"id":4906},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":618},"inlinerMath-9rSWhNw25-10",{"__TSPROSE_proseElement":175,"schema":4908,"data":4909},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hours:","paragraph-FVqaXxo0h",{"__TSPROSE_proseElement":175,"schema":4912,"data":4913,"storageKey":4915,"id":4916},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":4914,"freeze":246},"W = (P_1 + P_2) \\cdot 6","$$ W = (P_1 + P_2) \\cdot 6 $$","blockMath-yqlhsonTc",{"__TSPROSE_proseElement":175,"schema":4918,"children":4919,"id":4952},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4920,4923,4927,4930,4934,4937,4941,4944,4950],{"__TSPROSE_proseElement":175,"schema":4921,"data":4922},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"We are not given any rates, so replace them with expressions through ",{"__TSPROSE_proseElement":175,"schema":4924,"data":4925,"storageKey":4746,"id":4926},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4745},"inlinerMath-m6vjYMXNe-1",{"__TSPROSE_proseElement":175,"schema":4928,"data":4929},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and time. From ",{"__TSPROSE_proseElement":175,"schema":4931,"data":4932,"storageKey":4801,"id":4933},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4800},"inlinerMath-xPOEN3cy3-2",{"__TSPROSE_proseElement":175,"schema":4935,"data":4936},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", if you divide both sides by ",{"__TSPROSE_proseElement":175,"schema":4938,"data":4939,"storageKey":2236,"id":4940},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":2235},"inlinerMath-NdU0V7CGr-2",{"__TSPROSE_proseElement":175,"schema":4942,"data":4943},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", you get ",{"__TSPROSE_proseElement":175,"schema":4945,"data":4946,"storageKey":4948,"id":4949},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4947},"P = \\frac{W}{t}","$ P = \\frac{W}{t} $","inlinerMath-flaS1pcwJ",{"__TSPROSE_proseElement":175,"schema":4951,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-b5MY9BwwB",{"__TSPROSE_proseElement":175,"schema":4954,"data":4955,"storageKey":4957,"id":4958},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":4956,"freeze":246},"W = \\left(\\frac{W}{t_1} + \\frac{W}{t_2}\\right) \\cdot 6","$$ W = \\left(\\frac{W}{t_1} + \\frac{W}{t_2}\\right) \\cdot 6 $$","blockMath-11su9RGMR",{"__TSPROSE_proseElement":175,"schema":4960,"children":4961,"id":4972},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4962,4965,4969],{"__TSPROSE_proseElement":175,"schema":4963,"data":4964},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Factor out ",{"__TSPROSE_proseElement":175,"schema":4966,"data":4967,"storageKey":4746,"id":4968},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4745},"inlinerMath-m6vjYMXNe-2",{"__TSPROSE_proseElement":175,"schema":4970,"data":4971},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and divide both sides by it. That completely kills the total-work quantity, whatever you imagine it to be:","paragraph-KSfHUmB78",{"__TSPROSE_proseElement":175,"schema":4974,"data":4975,"storageKey":4977,"id":4978},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":4976,"freeze":246},"\\cancel{W} = \\cancel{W} \\cdot \\left(\\frac{1}{t_1} + \\frac{1}{t_2}\\right) \\cdot 6 \\\\ 1 = \\left(\\frac{1}{t_1} + \\frac{1}{t_2}\\right) \\cdot 6 \\\\ \\frac{1}{6} = \\frac{1}{t_1} + \\frac{1}{t_2}","$$ \\cancel{W} = \\cancel{W} \\cdot \\left(\\frac{1}{t_1} + \\frac{1}{t_2}\\right) \\cdot 6 \\\\ 1 = \\left(\\frac{1}{t_1} + \\frac{1}{t_2}\\right) \\cdot 6 \\\\ \\frac{1}{6} = \\frac{1}{t_1} + \\frac{1}{t_2} $$","blockMath-EsfV0EX3k",{"__TSPROSE_proseElement":175,"schema":4980,"children":4981,"id":5000},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[4982,4985,4989,4992,4998],{"__TSPROSE_proseElement":175,"schema":4983,"data":4984},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Nice: now the equation contains only time, which is exactly what we need. We also know the first worker is ",{"__TSPROSE_proseElement":175,"schema":4986,"data":4987,"storageKey":796,"id":4988},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":795},"inlinerMath-MAD3UaxRg-5",{"__TSPROSE_proseElement":175,"schema":4990,"data":4991},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hours faster than the second, so ",{"__TSPROSE_proseElement":175,"schema":4993,"data":4994,"storageKey":4996,"id":4997},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4995},"t_1 = t_2 - 5","$ t_1 = t_2 - 5 $","inlinerMath-zJEWtPi6T",{"__TSPROSE_proseElement":175,"schema":4999,"data":1977},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-SAvJcMxDE",{"__TSPROSE_proseElement":175,"schema":5002,"data":5003,"storageKey":5005,"id":5006},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5004,"freeze":246},"\\frac{1}{6} = \\frac{1}{t_2 - 5} + \\frac{1}{t_2}","$$ \\frac{1}{6} = \\frac{1}{t_2 - 5} + \\frac{1}{t_2} $$","blockMath-sWb36ANLW",{"__TSPROSE_proseElement":175,"schema":5008,"children":5009,"id":5013},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5010],{"__TSPROSE_proseElement":175,"schema":5011,"data":5012},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Now solve it:","paragraph-LNQW1dKw9",{"__TSPROSE_proseElement":175,"schema":5015,"data":5016,"storageKey":5018,"id":5019},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5017,"freeze":246},"\\frac{(t_2 - 5)t_2}{6} = t_2 + t_2 - 5 \\\\ \\frac{t_2^2 - 5t_2}{6} = 2t_2 - 5 \\\\ t_2^2 - 5t_2 = 12t_2 - 30 \\\\ t_2^2 - 17t_2 + 30 = 0","$$ \\frac{(t_2 - 5)t_2}{6} = t_2 + t_2 - 5 \\\\ \\frac{t_2^2 - 5t_2}{6} = 2t_2 - 5 \\\\ t_2^2 - 5t_2 = 12t_2 - 30 \\\\ t_2^2 - 17t_2 + 30 = 0 $$","blockMath-FT823xSkP",{"__TSPROSE_proseElement":175,"schema":5021,"children":5022,"id":5069},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5023,5026,5030,5033,5037,5040,5046,5048,5052,5055,5061,5063,5067],{"__TSPROSE_proseElement":175,"schema":5024,"data":5025},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Yep, a quadratic equation. Surprise. Since the leading coefficient is ",{"__TSPROSE_proseElement":175,"schema":5027,"data":5028,"storageKey":1631,"id":5029},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1630},"inlinerMath-GWJ3eIxvK-9",{"__TSPROSE_proseElement":175,"schema":5031,"data":5032},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and the other numbers are simple, you can ",{"__TSPROSE_proseElement":175,"schema":5034,"data":5035,"storageKey":496,"id":5036},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":495},"dependencyInliner-fdi2FzGId-4",{"__TSPROSE_proseElement":175,"schema":5038,"data":5039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". 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So the only valid option is ",{"__TSPROSE_proseElement":175,"schema":5112,"data":5113,"storageKey":1785,"id":5114},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1784},"inlinerMath-1xC295UiR-4",{"__TSPROSE_proseElement":175,"schema":5116,"data":5117},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hours for the second worker and therefore ",{"__TSPROSE_proseElement":175,"schema":5119,"data":5120,"storageKey":1085,"id":5121},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1084},"inlinerMath-8Y3Z60Vbg-8",{"__TSPROSE_proseElement":175,"schema":5123,"data":5124},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hours for the first.","paragraph-XFyubp261","painting-a-fence",{"__TSPROSE_proseElement":175,"schema":5128,"data":5129,"children":5133,"id":5393},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":5130},{"title":5131,"level":1141,"attributes":5132},"Plowing a Field",[],[5134,5176,5187,5197,5222],{"__TSPROSE_proseElement":175,"schema":5135,"children":5136},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[5137],{"__TSPROSE_proseElement":175,"schema":5138,"children":5139,"id":5175},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5140,5143,5149,5152,5158,5161,5165,5168,5172],{"__TSPROSE_proseElement":175,"schema":5141,"data":5142},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"One tractor brigade plowed ",{"__TSPROSE_proseElement":175,"schema":5144,"data":5145,"storageKey":5147,"id":5148},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5146},"240","$ 240 $","inlinerMath-7jme5924B",{"__TSPROSE_proseElement":175,"schema":5150,"data":5151},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hectares, while another plowed ",{"__TSPROSE_proseElement":175,"schema":5153,"data":5154,"storageKey":5156,"id":5157},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5155},"35%","$ 35% $","inlinerMath-U4vJcm9D0",{"__TSPROSE_proseElement":175,"schema":5159,"data":5160},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," more. 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How many hectares per day did each brigade plow, given that both exceeded the planned daily quota of ",{"__TSPROSE_proseElement":175,"schema":5169,"data":5170,"storageKey":1166,"id":5171},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1165},"inlinerMath-pnamxzA0J-1",{"__TSPROSE_proseElement":175,"schema":5173,"data":5174},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hectares?","paragraph-2W9bw27jW",{"__TSPROSE_proseElement":175,"schema":5177,"data":5178},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":5179,"serializedValidator":5180},"Brigade work rates",{"type":1177,"ordered":246,"separator":1178,"answers":5181},[5182,5184],{"__ERUDIT_CHECK":175,"name":396,"data":5183},{"expr":4457},{"__ERUDIT_CHECK":175,"name":396,"data":5185},{"expr":5186},"27",{"__TSPROSE_proseElement":175,"schema":5188,"children":5189},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[5190],{"__TSPROSE_proseElement":175,"schema":5191,"children":5192,"id":5196},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5193],{"__TSPROSE_proseElement":175,"schema":5194,"data":5195},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Use the relation between the work times of the two brigades. Replace time with expressions through work amount and work rate.","paragraph-qmLjkoLVI",{"__TSPROSE_proseElement":175,"schema":5198,"children":5199},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[5200],{"__TSPROSE_proseElement":175,"schema":5201,"children":5202,"id":5221},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5203,5206,5210,5213,5218],{"__TSPROSE_proseElement":175,"schema":5204,"data":5205},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The first brigade plowed ",{"__TSPROSE_proseElement":175,"schema":5207,"data":5208,"storageKey":4488,"id":5209},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4457},"inlinerMath-vVUmnH6Sj-2",{"__TSPROSE_proseElement":175,"schema":5211,"data":5212},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hectares per day, and the second plowed ",{"__TSPROSE_proseElement":175,"schema":5214,"data":5215,"storageKey":5216,"id":5217},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5186},"$ 27 $","inlinerMath-IW9FMi3Jk",{"__TSPROSE_proseElement":175,"schema":5219,"data":5220},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hectares per day.","paragraph-qmVXYKpFS",{"__TSPROSE_proseElement":175,"schema":5223,"children":5224},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[5225,5241,5247,5254,5260,5267,5273,5297,5303,5316,5322,5335,5341],{"__TSPROSE_proseElement":175,"schema":5226,"children":5227,"id":5240},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5228,5231,5237],{"__TSPROSE_proseElement":175,"schema":5229,"data":5230},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Translate each sentence directly into math. First, the first brigade did ",{"__TSPROSE_proseElement":175,"schema":5232,"data":5233,"storageKey":5235,"id":5236},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5234},"W_1 = 240","$ W_1 = 240 $","inlinerMath-9Jk4r3i41",{"__TSPROSE_proseElement":175,"schema":5238,"data":5239},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hectares of work. From that, find the second brigade's amount immediately:","paragraph-VRYY1VycB",{"__TSPROSE_proseElement":175,"schema":5242,"data":5243,"storageKey":5245,"id":5246},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5244,"freeze":246},"W_2 = W_1 + \\frac{35}{100} \\cdot W_1 = 240 + \\frac{35}{100} \\cdot 240 = 324","$$ W_2 = W_1 + \\frac{35}{100} \\cdot W_1 = 240 + \\frac{35}{100} \\cdot 240 = 324 $$","blockMath-BNW67XsAn",{"__TSPROSE_proseElement":175,"schema":5248,"children":5249,"id":5253},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5250],{"__TSPROSE_proseElement":175,"schema":5251,"data":5252},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The second sentence gives the relation between daily rates and completion times. Translate that:","paragraph-5PcCvP0w1",{"__TSPROSE_proseElement":175,"schema":5255,"data":5256,"storageKey":5258,"id":5259},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5257,"freeze":246},"P_1 = P_2 - 3 >>{big} t_1 = t_2 - 2","$$ P_1 = P_2 - 3 >>{big} t_1 = t_2 - 2 $$","blockMath-q2VVSWXHM",{"__TSPROSE_proseElement":175,"schema":5261,"children":5262,"id":5266},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5263],{"__TSPROSE_proseElement":175,"schema":5264,"data":5265},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Either relation could be used as the main one, but since the task asks for the rates, start from the time relation and replace time by expressions through work amount and work rate:","paragraph-B9agRHKkl",{"__TSPROSE_proseElement":175,"schema":5268,"data":5269,"storageKey":5271,"id":5272},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5270,"freeze":246},"\\frac{W_1}{P_1} = \\frac{W_2}{P_2} - 2","$$ \\frac{W_1}{P_1} = \\frac{W_2}{P_2} - 2 $$","blockMath-vp6g5XKSB",{"__TSPROSE_proseElement":175,"schema":5274,"children":5275,"id":5296},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5276,5279,5285,5287,5293],{"__TSPROSE_proseElement":175,"schema":5277,"data":5278},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Now substitute the expression for ",{"__TSPROSE_proseElement":175,"schema":5280,"data":5281,"storageKey":5283,"id":5284},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5282},"P_1","$ P_1 $","inlinerMath-hvGY9z42D",{"__TSPROSE_proseElement":175,"schema":5286,"data":1961},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5288,"data":5289,"storageKey":5291,"id":5292},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5290},"P_2","$ P_2 $","inlinerMath-UcV5Mm3dY",{"__TSPROSE_proseElement":175,"schema":5294,"data":5295},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," from the first relation, and also the numeric work amounts:","paragraph-umfjmozDS",{"__TSPROSE_proseElement":175,"schema":5298,"data":5299,"storageKey":5301,"id":5302},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5300,"freeze":246},"\\frac{240}{P_2 - 3} = \\frac{324}{P_2} - 2 \\\\ 240P_2 = 324(P_2 - 3) - 2P_2(P_2 - 3) \\\\ 120P_2 = 162(P_2 - 3) - P_2(P_2 - 3) \\\\ 120P_2 = 162P_2 - 486 - P_2^2 + 3P_2 \\\\ P_2^2 - 45P_2 + 486 = 0","$$ \\frac{240}{P_2 - 3} = \\frac{324}{P_2} - 2 \\\\ 240P_2 = 324(P_2 - 3) - 2P_2(P_2 - 3) \\\\ 120P_2 = 162(P_2 - 3) - P_2(P_2 - 3) \\\\ 120P_2 = 162P_2 - 486 - P_2^2 + 3P_2 \\\\ P_2^2 - 45P_2 + 486 = 0 $$","blockMath-EYBdS4D5G",{"__TSPROSE_proseElement":175,"schema":5304,"children":5305,"id":5315},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5306,5309,5313],{"__TSPROSE_proseElement":175,"schema":5307,"data":5308},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"We got a quadratic equation. Find the ",{"__TSPROSE_proseElement":175,"schema":5310,"data":5311,"storageKey":2427,"id":5312},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":193},"dependencyInliner-emFvGyx3M-3",{"__TSPROSE_proseElement":175,"schema":5314,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-60rl0yBXm",{"__TSPROSE_proseElement":175,"schema":5317,"data":5318,"storageKey":5320,"id":5321},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5319,"freeze":246},"D = (-45)^2 - 4 \\cdot 1 \\cdot 486 = 2025 - 1944 = 81","$$ D = (-45)^2 - 4 \\cdot 1 \\cdot 486 = 2025 - 1944 = 81 $$","blockMath-QYHhsA97U",{"__TSPROSE_proseElement":175,"schema":5323,"children":5324,"id":5334},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5325,5328,5332],{"__TSPROSE_proseElement":175,"schema":5326,"data":5327},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Then the roots using the ",{"__TSPROSE_proseElement":175,"schema":5329,"data":5330,"storageKey":2448,"id":5331},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":3258},"dependencyInliner-1X2UJbegR-1",{"__TSPROSE_proseElement":175,"schema":5333,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-9ZmED7uqh",{"__TSPROSE_proseElement":175,"schema":5336,"data":5337,"storageKey":5339,"id":5340},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5338,"freeze":246},"P_2 = \\frac{-(-45) \\pm \\sqrt{81}}{2 \\cdot 1} = \\frac{45 \\pm 9}{2}","$$ P_2 = \\frac{-(-45) \\pm \\sqrt{81}}{2 \\cdot 1} = \\frac{45 \\pm 9}{2} $$","blockMath-KsflWUvv2",{"__TSPROSE_proseElement":175,"schema":5342,"children":5343,"id":5392},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5344,5346,5352,5354,5360,5363,5367,5370,5374,5377,5381,5384,5390],{"__TSPROSE_proseElement":175,"schema":5345,"data":2593},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5347,"data":5348,"storageKey":5350,"id":5351},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5349},"P_2 = 27","$ P_2 = 27 $","inlinerMath-e2712Cpbw",{"__TSPROSE_proseElement":175,"schema":5353,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5355,"data":5356,"storageKey":5358,"id":5359},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5357},"P_2 = 18","$ P_2 = 18 $","inlinerMath-kUpHO4Dre",{"__TSPROSE_proseElement":175,"schema":5361,"data":5362},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". But both brigades are said to exceed the ",{"__TSPROSE_proseElement":175,"schema":5364,"data":5365,"storageKey":1166,"id":5366},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1165},"inlinerMath-pnamxzA0J-2",{"__TSPROSE_proseElement":175,"schema":5368,"data":5369},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"-hectare quota, so the second brigade's rate cannot be below ",{"__TSPROSE_proseElement":175,"schema":5371,"data":5372,"storageKey":1166,"id":5373},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1165},"inlinerMath-pnamxzA0J-3",{"__TSPROSE_proseElement":175,"schema":5375,"data":5376},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Therefore the only valid option is ",{"__TSPROSE_proseElement":175,"schema":5378,"data":5379,"storageKey":5350,"id":5380},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5349},"inlinerMath-e2712Cpbw-1",{"__TSPROSE_proseElement":175,"schema":5382,"data":5383},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," hectares per day, which means ",{"__TSPROSE_proseElement":175,"schema":5385,"data":5386,"storageKey":5388,"id":5389},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5387},"P_1 = 27 - 3 = 24","$ P_1 = 27 - 3 = 24 $","inlinerMath-cA19QOTTg",{"__TSPROSE_proseElement":175,"schema":5391,"data":5220},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-UQrxOkHJi","plowing-a-field",{"__TSPROSE_proseElement":175,"schema":5395,"data":5396,"children":5400,"id":5717},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":5397},{"title":5398,"level":1419,"attributes":5399},"3 Taps -- 1 Tank",[],[5401,5434,5447,5457,5490],{"__TSPROSE_proseElement":175,"schema":5402,"children":5403},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[5404],{"__TSPROSE_proseElement":175,"schema":5405,"children":5406,"id":5433},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5407,5410,5414,5417,5423,5426,5430],{"__TSPROSE_proseElement":175,"schema":5408,"data":5409},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"A liquid flows into a tank through three inlet taps. If all three taps are opened at once, the tank fills in ",{"__TSPROSE_proseElement":175,"schema":5411,"data":5412,"storageKey":619,"id":5413},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":618},"inlinerMath-9rSWhNw25-11",{"__TSPROSE_proseElement":175,"schema":5415,"data":5416},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," minutes. If only the second tap is used, the tank fills in ",{"__TSPROSE_proseElement":175,"schema":5418,"data":5419,"storageKey":5421,"id":5422},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5420},"3\u002F4","$ 3\u002F4 $","inlinerMath-7pQOp7GKd",{"__TSPROSE_proseElement":175,"schema":5424,"data":5425},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," of the time needed by the first tap alone. The third tap alone fills the tank in ",{"__TSPROSE_proseElement":175,"schema":5427,"data":5428,"storageKey":1085,"id":5429},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1084},"inlinerMath-8Y3Z60Vbg-9",{"__TSPROSE_proseElement":175,"schema":5431,"data":5432},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," minutes longer than the second tap alone. How long does each tap take to fill the tank by itself?","paragraph-katJOm77Q",{"__TSPROSE_proseElement":175,"schema":5435,"data":5436},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":5437,"serializedValidator":5438},"Fill time of each tap",{"type":1177,"ordered":246,"separator":1178,"answers":5439},[5440,5443,5445],{"__ERUDIT_CHECK":175,"name":396,"data":5441},{"expr":5442},"56\u002F3",{"__ERUDIT_CHECK":175,"name":396,"data":5444},{"expr":3389},{"__ERUDIT_CHECK":175,"name":396,"data":5446},{"expr":4457},{"__TSPROSE_proseElement":175,"schema":5448,"children":5449},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[5450],{"__TSPROSE_proseElement":175,"schema":5451,"children":5452,"id":5456},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5453],{"__TSPROSE_proseElement":175,"schema":5454,"data":5455},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Start from the work formula for the full tank when all taps are open. In the sum of rates, replace each rate by an expression through total work and fill time.","paragraph-NRCQXBwY5",{"__TSPROSE_proseElement":175,"schema":5458,"children":5459},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[5460],{"__TSPROSE_proseElement":175,"schema":5461,"children":5462,"id":5489},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5463,5466,5472,5475,5479,5482,5486],{"__TSPROSE_proseElement":175,"schema":5464,"data":5465},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The first tap fills the tank in ",{"__TSPROSE_proseElement":175,"schema":5467,"data":5468,"storageKey":5470,"id":5471},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5469},"\\frac{56}{3}","$ \\frac{56}{3} $","inlinerMath-3LOq1XIcj",{"__TSPROSE_proseElement":175,"schema":5473,"data":5474},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," minutes, the second in ",{"__TSPROSE_proseElement":175,"schema":5476,"data":5477,"storageKey":3390,"id":5478},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3389},"inlinerMath-EG1ZwIV76-1",{"__TSPROSE_proseElement":175,"schema":5480,"data":5481},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," minutes, and the third in ",{"__TSPROSE_proseElement":175,"schema":5483,"data":5484,"storageKey":4488,"id":5485},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4457},"inlinerMath-vVUmnH6Sj-3",{"__TSPROSE_proseElement":175,"schema":5487,"data":5488},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," minutes.","paragraph-Qqhj2a7ha",{"__TSPROSE_proseElement":175,"schema":5491,"children":5492},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[5493,5500,5506,5513,5519,5526,5532,5545,5551,5565,5571,5593,5599,5614,5620,5632,5638,5650,5656,5686,5692],{"__TSPROSE_proseElement":175,"schema":5494,"children":5495,"id":5499},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5496],{"__TSPROSE_proseElement":175,"schema":5497,"data":5498},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Write the full-tank work formula for all three taps together:","paragraph-ZhC0dZWGT",{"__TSPROSE_proseElement":175,"schema":5501,"data":5502,"storageKey":5504,"id":5505},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5503,"freeze":246},"W = (P_1 + P_2 + P_3) \\cdot 6","$$ W = (P_1 + P_2 + P_3) \\cdot 6 $$","blockMath-PLKAdj2I9",{"__TSPROSE_proseElement":175,"schema":5507,"children":5508,"id":5512},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5509],{"__TSPROSE_proseElement":175,"schema":5510,"data":5511},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Now write the time relations:","paragraph-D1UCZrIaR",{"__TSPROSE_proseElement":175,"schema":5514,"data":5515,"storageKey":5517,"id":5518},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5516,"freeze":246},"t_2 = \\frac{3}{4}t_1 >>{big} t_3 = t_2 + 10","$$ t_2 = \\frac{3}{4}t_1 >>{big} t_3 = t_2 + 10 $$","blockMath-U2hHGi5fM",{"__TSPROSE_proseElement":175,"schema":5520,"children":5521,"id":5525},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5522],{"__TSPROSE_proseElement":175,"schema":5523,"data":5524},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Substitute expressions through work and time for the rates in the first formula, because time is what we need:","paragraph-kizNkiFZ0",{"__TSPROSE_proseElement":175,"schema":5527,"data":5528,"storageKey":5530,"id":5531},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5529,"freeze":246},"W = \\left(\\frac{W}{t_1} + \\frac{W}{t_2} + \\frac{W}{t_3}\\right) \\cdot 6","$$ W = \\left(\\frac{W}{t_1} + \\frac{W}{t_2} + \\frac{W}{t_3}\\right) \\cdot 6 $$","blockMath-zjU6rvIc1",{"__TSPROSE_proseElement":175,"schema":5533,"children":5534,"id":5544},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5535,5537,5541],{"__TSPROSE_proseElement":175,"schema":5536,"data":4964},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5538,"data":5539,"storageKey":4746,"id":5540},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4745},"inlinerMath-m6vjYMXNe-3",{"__TSPROSE_proseElement":175,"schema":5542,"data":5543},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," on the right and divide both sides by it. That completely eliminates the total-work quantity:","paragraph-9l2SJ90IJ",{"__TSPROSE_proseElement":175,"schema":5546,"data":5547,"storageKey":5549,"id":5550},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5548,"freeze":246},"1 = \\left(\\frac{1}{t_1} + \\frac{1}{t_2} + \\frac{1}{t_3}\\right) \\cdot 6","$$ 1 = \\left(\\frac{1}{t_1} + \\frac{1}{t_2} + \\frac{1}{t_3}\\right) \\cdot 6 $$","blockMath-BnMSxTSAd",{"__TSPROSE_proseElement":175,"schema":5552,"children":5553,"id":5564},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5554,5557,5561],{"__TSPROSE_proseElement":175,"schema":5555,"data":5556},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Now express everything through a single time, for example ",{"__TSPROSE_proseElement":175,"schema":5558,"data":5559,"storageKey":4283,"id":5560},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4282},"inlinerMath-gIzVuylZa-1",{"__TSPROSE_proseElement":175,"schema":5562,"data":5563},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". From the first time relation we get:","paragraph-WWdXYYhle",{"__TSPROSE_proseElement":175,"schema":5566,"data":5567,"storageKey":5569,"id":5570},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5568,"freeze":246},"t_2 = \\frac{3}{4}t_1 \\implies t_1 = \\frac{4}{3}t_2","$$ t_2 = \\frac{3}{4}t_1 \\implies t_1 = \\frac{4}{3}t_2 $$","blockMath-QibhAe9ZM",{"__TSPROSE_proseElement":175,"schema":5572,"children":5573,"id":5592},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5574,5577,5581,5584,5590],{"__TSPROSE_proseElement":175,"schema":5575,"data":5576},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Substitute this for ",{"__TSPROSE_proseElement":175,"schema":5578,"data":5579,"storageKey":3742,"id":5580},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":3741},"inlinerMath-tbk2hBmyF-4",{"__TSPROSE_proseElement":175,"schema":5582,"data":5583},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and also ",{"__TSPROSE_proseElement":175,"schema":5585,"data":5586,"storageKey":5588,"id":5589},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5587},"t_3 = t_2 + 10","$ t_3 = t_2 + 10 $","inlinerMath-r9eeJ3ugV",{"__TSPROSE_proseElement":175,"schema":5591,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-Q0D5yYpvC",{"__TSPROSE_proseElement":175,"schema":5594,"data":5595,"storageKey":5597,"id":5598},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5596,"freeze":246},"1 = \\left(\\frac{1}{\\frac{4}{3}t_2} + \\frac{1}{t_2} + \\frac{1}{t_2 + 10}\\right) \\cdot 6 \\\\ \\frac{1}{6} = \\frac{3}{4t_2} + \\frac{1}{t_2} + \\frac{1}{t_2 + 10} \\\\ \\frac{1}{6} = \\frac{7}{4t_2} + \\frac{1}{t_2 + 10}","$$ 1 = \\left(\\frac{1}{\\frac{4}{3}t_2} + \\frac{1}{t_2} + \\frac{1}{t_2 + 10}\\right) \\cdot 6 \\\\ \\frac{1}{6} = \\frac{3}{4t_2} + \\frac{1}{t_2} + \\frac{1}{t_2 + 10} \\\\ \\frac{1}{6} = \\frac{7}{4t_2} + \\frac{1}{t_2 + 10} $$","blockMath-rh4AxvGKh",{"__TSPROSE_proseElement":175,"schema":5600,"children":5601,"id":5613},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5602,5604,5610],{"__TSPROSE_proseElement":175,"schema":5603,"data":1990},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5605,"data":5606,"storageKey":5608,"id":5609},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5607},"12t_2(t_2 + 10)","$ 12t_2(t_2 + 10) $","inlinerMath-hjGU31Mo3",{"__TSPROSE_proseElement":175,"schema":5611,"data":5612},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," to clear all denominators:","paragraph-2Q7POYBBi",{"__TSPROSE_proseElement":175,"schema":5615,"data":5616,"storageKey":5618,"id":5619},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5617,"freeze":246},"2t_2(t_2 + 10) = 21(t_2 + 10) + 12t_2 \\\\ 2t_2^2 + 20t_2 = 21t_2 + 210 + 12t_2 \\\\ 2t_2^2 - 13t_2 - 210 = 0","$$ 2t_2(t_2 + 10) = 21(t_2 + 10) + 12t_2 \\\\ 2t_2^2 + 20t_2 = 21t_2 + 210 + 12t_2 \\\\ 2t_2^2 - 13t_2 - 210 = 0 $$","blockMath-SGF7W2hfe",{"__TSPROSE_proseElement":175,"schema":5621,"children":5622,"id":5631},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5623,5625,5629],{"__TSPROSE_proseElement":175,"schema":5624,"data":2423},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5626,"data":5627,"storageKey":2427,"id":5628},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":193},"dependencyInliner-emFvGyx3M-4",{"__TSPROSE_proseElement":175,"schema":5630,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-pzCqNZgqf-1",{"__TSPROSE_proseElement":175,"schema":5633,"data":5634,"storageKey":5636,"id":5637},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5635,"freeze":246},"D = (-13)^2 - 4 \\cdot 2 \\cdot (-210) = 169 + 1680 = 1849","$$ D = (-13)^2 - 4 \\cdot 2 \\cdot (-210) = 169 + 1680 = 1849 $$","blockMath-wC9pY2HzS",{"__TSPROSE_proseElement":175,"schema":5639,"children":5640,"id":5649},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5641,5643,5647],{"__TSPROSE_proseElement":175,"schema":5642,"data":5327},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5644,"data":5645,"storageKey":2448,"id":5646},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":3258},"dependencyInliner-1X2UJbegR-2",{"__TSPROSE_proseElement":175,"schema":5648,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-9ZmED7uqh-1",{"__TSPROSE_proseElement":175,"schema":5651,"data":5652,"storageKey":5654,"id":5655},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5653,"freeze":246},"t_2 = \\frac{-(-13) \\pm \\sqrt{1849}}{2 \\cdot 2} = \\frac{13 \\pm 43}{4}","$$ t_2 = \\frac{-(-13) \\pm \\sqrt{1849}}{2 \\cdot 2} = \\frac{13 \\pm 43}{4} $$","blockMath-siZhAG5Vq",{"__TSPROSE_proseElement":175,"schema":5657,"children":5658,"id":5685},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5659,5661,5667,5669,5675,5678,5682],{"__TSPROSE_proseElement":175,"schema":5660,"data":2593},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5662,"data":5663,"storageKey":5665,"id":5666},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5664},"t_2 = 14","$ t_2 = 14 $","inlinerMath-qQrNM4bji",{"__TSPROSE_proseElement":175,"schema":5668,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5670,"data":5671,"storageKey":5673,"id":5674},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5672},"t_2 = -7.5","$ t_2 = -7.5 $","inlinerMath-0AlLQuvXc",{"__TSPROSE_proseElement":175,"schema":5676,"data":5677},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Time cannot be negative, so ",{"__TSPROSE_proseElement":175,"schema":5679,"data":5680,"storageKey":5665,"id":5681},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5664},"inlinerMath-qQrNM4bji-1",{"__TSPROSE_proseElement":175,"schema":5683,"data":5684},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," minutes. 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In English, ",{"__TSPROSE_proseElement":175,"schema":5730,"data":5731,"children":5732,"id":5736},{"name":274,"type":255,"linkable":175,"__TSPROSE_schema":175},{"type":276},[5733],{"__TSPROSE_proseElement":175,"schema":5734,"data":5735},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"mixture","emphasis-gCbdSmhZ0",{"__TSPROSE_proseElement":175,"schema":5738,"data":5739},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is the broad general word,",{"__TSPROSE_proseElement":175,"schema":5741,"data":5742,"children":5743,"id":5747},{"name":274,"type":255,"linkable":175,"__TSPROSE_schema":175},{"type":276},[5744],{"__TSPROSE_proseElement":175,"schema":5745,"data":5746},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"solution","emphasis-YYRFcsd1e",{"__TSPROSE_proseElement":175,"schema":5749,"data":5750},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is used when one substance is dissolved in another, and ",{"__TSPROSE_proseElement":175,"schema":5752,"data":5753,"children":5754,"id":5758},{"name":274,"type":255,"linkable":175,"__TSPROSE_schema":175},{"type":276},[5755],{"__TSPROSE_proseElement":175,"schema":5756,"data":5757},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"alloy","emphasis-kStFUyF1K",{"__TSPROSE_proseElement":175,"schema":5760,"data":5761},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is the standard word for a metallic mixture. So in this section we will look at a liquid solution, a metal alloy, and a gas mixture.","paragraph-9TsazRWUP",{"__TSPROSE_proseElement":175,"schema":5764,"children":5765,"id":5806},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5766,5769,5777,5780,5788,5791,5795,5798,5804],{"__TSPROSE_proseElement":175,"schema":5767,"data":5768},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"For all these systems, it is crucial to understand ",{"__TSPROSE_proseElement":175,"schema":5770,"data":5771,"children":5772,"id":5776},{"name":274,"type":255,"linkable":175,"__TSPROSE_schema":175},{"type":276},[5773],{"__TSPROSE_proseElement":175,"schema":5774,"data":5775},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"what fraction","emphasis-ZFKqTN0SD",{"__TSPROSE_proseElement":175,"schema":5778,"data":5779},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," of the total mass or volume is made up by the substance you care about. For example, what fraction of an acid-and-water solution is acid. That ratio is called the ",{"__TSPROSE_proseElement":175,"schema":5781,"data":5782,"children":5783,"id":5787},{"name":274,"type":255,"linkable":175,"__TSPROSE_schema":175},{"type":276},[5784],{"__TSPROSE_proseElement":175,"schema":5785,"data":5786},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"concentration","emphasis-VTsebhusp",{"__TSPROSE_proseElement":175,"schema":5789,"data":5790},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," of the substance, say ",{"__TSPROSE_proseElement":175,"schema":5792,"data":5793,"storageKey":1622,"id":5794},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1621},"inlinerMath-eIvv0Olnw-1",{"__TSPROSE_proseElement":175,"schema":5796,"data":5797},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", and it is denoted by ",{"__TSPROSE_proseElement":175,"schema":5799,"data":5800,"storageKey":5802,"id":5803},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5801},"c","$ c $","inlinerMath-8Adf0APnu",{"__TSPROSE_proseElement":175,"schema":5805,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-G2vdQmzZ0",{"__TSPROSE_proseElement":175,"schema":5808,"data":5809,"storageKey":5811,"id":5812},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5810,"freeze":246},"c_A = \\frac{m_A}{M} >>{big} c_A = \\frac{V_A}{V}","$$ c_A = \\frac{m_A}{M} >>{big} c_A = \\frac{V_A}{V} $$","blockMath-rZwQdFKGp",{"__TSPROSE_proseElement":175,"schema":5814,"children":5815,"id":5836},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5816,5819,5825,5828,5834],{"__TSPROSE_proseElement":175,"schema":5817,"data":5818},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Concentration is usually written in percent, but in calculations you use decimals. For example, a hydrogen peroxide solution with concentration ",{"__TSPROSE_proseElement":175,"schema":5820,"data":5821,"storageKey":5823,"id":5824},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5822},"3%","$ 3% $","inlinerMath-xbaaV1sW1",{"__TSPROSE_proseElement":175,"schema":5826,"data":5827},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," means ",{"__TSPROSE_proseElement":175,"schema":5829,"data":5830,"storageKey":5832,"id":5833},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5831},"c = 0.03","$ c = 0.03 $","inlinerMath-SodLSQxx8",{"__TSPROSE_proseElement":175,"schema":5835,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-MenjSuL1k",{"__TSPROSE_proseElement":175,"schema":5838,"data":5839,"children":5843,"id":6074},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":5840},{"title":5841,"level":341,"attributes":5842},"Salt Solution",[],[5844,5879,5892,5902,5927],{"__TSPROSE_proseElement":175,"schema":5845,"children":5846},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[5847],{"__TSPROSE_proseElement":175,"schema":5848,"children":5849,"id":5878},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5850,5853,5857,5860,5866,5869,5875],{"__TSPROSE_proseElement":175,"schema":5851,"data":5852},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"A solution containing ",{"__TSPROSE_proseElement":175,"schema":5854,"data":5855,"storageKey":4480,"id":5856},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":4454},"inlinerMath-OyvVmaRZW-2",{"__TSPROSE_proseElement":175,"schema":5858,"data":5859},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," grams of salt has another ",{"__TSPROSE_proseElement":175,"schema":5861,"data":5862,"storageKey":5864,"id":5865},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5863},"600","$ 600 $","inlinerMath-Npc6nf023",{"__TSPROSE_proseElement":175,"schema":5867,"data":5868},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," grams of water added to it, after which the concentration of the solution decreases by ",{"__TSPROSE_proseElement":175,"schema":5870,"data":5871,"storageKey":5873,"id":5874},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5872},"4%","$ 4% $","inlinerMath-aq3nI0Xve",{"__TSPROSE_proseElement":175,"schema":5876,"data":5877},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Find the original salt concentration.","paragraph-sCPXogzTA",{"__TSPROSE_proseElement":175,"schema":5880,"data":5881},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":5882,"serializedValidator":5883},"Original salt concentration",{"type":5884,"answer":5885},"value",[5886,5889],{"__ERUDIT_CHECK":175,"name":396,"data":5887},{"expr":5888},"0.06",{"__ERUDIT_CHECK":175,"name":396,"data":5890},{"expr":5891},"6%",{"__TSPROSE_proseElement":175,"schema":5893,"children":5894},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[5895],{"__TSPROSE_proseElement":175,"schema":5896,"children":5897,"id":5901},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5898],{"__TSPROSE_proseElement":175,"schema":5899,"data":5900},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Write the concentration formula before and after adding the water. In the second formula, substitute the expression for the original concentration.","paragraph-NpYEXd0Cf",{"__TSPROSE_proseElement":175,"schema":5903,"children":5904},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[5905],{"__TSPROSE_proseElement":175,"schema":5906,"children":5907,"id":5926},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5908,5911,5916,5919,5924],{"__TSPROSE_proseElement":175,"schema":5909,"data":5910},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The original salt concentration was ",{"__TSPROSE_proseElement":175,"schema":5912,"data":5913,"storageKey":5914,"id":5915},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5888},"$ 0.06 $","inlinerMath-hh823ZCMd",{"__TSPROSE_proseElement":175,"schema":5917,"data":5918},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", or ",{"__TSPROSE_proseElement":175,"schema":5920,"data":5921,"storageKey":5922,"id":5923},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5891},"$ 6% $","inlinerMath-yK3Dwyjbq",{"__TSPROSE_proseElement":175,"schema":5925,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-9ZEUBPkEt",{"__TSPROSE_proseElement":175,"schema":5928,"children":5929},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[5930,5952,5958,5987,5993,6006,6012,6032,6038,6068],{"__TSPROSE_proseElement":175,"schema":5931,"children":5932,"id":5951},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5933,5935,5941,5944,5948],{"__TSPROSE_proseElement":175,"schema":5934,"data":1869},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5936,"data":5937,"storageKey":5939,"id":5940},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5938},"M","$ M $","inlinerMath-0MhrSCdD8",{"__TSPROSE_proseElement":175,"schema":5942,"data":5943},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," be the mass of the original solution and ",{"__TSPROSE_proseElement":175,"schema":5945,"data":5946,"storageKey":5802,"id":5947},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5801},"inlinerMath-8Adf0APnu-1",{"__TSPROSE_proseElement":175,"schema":5949,"data":5950},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," its concentration. Then mathematically:","paragraph-DLQqvjgsh",{"__TSPROSE_proseElement":175,"schema":5953,"data":5954,"storageKey":5956,"id":5957},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5955,"freeze":246},"c = \\frac{18}{M}","$$ c = \\frac{18}{M} $$","blockMath-7nqU176mt",{"__TSPROSE_proseElement":175,"schema":5959,"children":5960,"id":5986},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5961,5964,5968,5970,5974,5977,5983],{"__TSPROSE_proseElement":175,"schema":5962,"data":5963},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"If we reduce the concentration ",{"__TSPROSE_proseElement":175,"schema":5965,"data":5966,"storageKey":5802,"id":5967},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5801},"inlinerMath-8Adf0APnu-2",{"__TSPROSE_proseElement":175,"schema":5969,"data":884},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":5971,"data":5972,"storageKey":5873,"id":5973},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5872},"inlinerMath-aq3nI0Xve-1",{"__TSPROSE_proseElement":175,"schema":5975,"data":5976},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", that is by ",{"__TSPROSE_proseElement":175,"schema":5978,"data":5979,"storageKey":5981,"id":5982},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5980},"0.04","$ 0.04 $","inlinerMath-BNi6dvEAx",{"__TSPROSE_proseElement":175,"schema":5984,"data":5985},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", we get the concentration of the diluted solution:","paragraph-HRKQx5mGD",{"__TSPROSE_proseElement":175,"schema":5988,"data":5989,"storageKey":5991,"id":5992},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":5990,"freeze":246},"c - 0.04 = \\frac{18}{M + 600}","$$ c - 0.04 = \\frac{18}{M + 600} $$","blockMath-n78g7jYMD",{"__TSPROSE_proseElement":175,"schema":5994,"children":5995,"id":6005},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[5996,5999,6003],{"__TSPROSE_proseElement":175,"schema":5997,"data":5998},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Substitute the fraction expression for ",{"__TSPROSE_proseElement":175,"schema":6000,"data":6001,"storageKey":5802,"id":6002},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5801},"inlinerMath-8Adf0APnu-3",{"__TSPROSE_proseElement":175,"schema":6004,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-n560jJaYt",{"__TSPROSE_proseElement":175,"schema":6007,"data":6008,"storageKey":6010,"id":6011},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6009,"freeze":246},"\\frac{18}{M} - 0.04 = \\frac{18}{M + 600} \\\\ \\frac{18}{M} - \\frac{4}{100} = \\frac{18}{M + 600} \\\\ \\frac{9}{M} - \\frac{2}{100} = \\frac{9}{M + 600} \\\\ 900(M+600) - 2M(M+600) = 900M \\\\ 450(M+600) - M(M+600) = 450M \\\\ \\cancel{450M} + 450 \\cdot 600 - M^2 - 600M = \\cancel{450M} \\\\ 450 \\cdot 600 - M^2 - 600M = 0 \\\\ M^2 + 600M - 450 \\cdot 600 = 0","$$ \\frac{18}{M} - 0.04 = \\frac{18}{M + 600} \\\\ \\frac{18}{M} - \\frac{4}{100} = \\frac{18}{M + 600} \\\\ \\frac{9}{M} - \\frac{2}{100} = \\frac{9}{M + 600} \\\\ 900(M+600) - 2M(M+600) = 900M \\\\ 450(M+600) - M(M+600) = 450M \\\\ \\cancel{450M} + 450 \\cdot 600 - M^2 - 600M = \\cancel{450M} \\\\ 450 \\cdot 600 - M^2 - 600M = 0 \\\\ M^2 + 600M - 450 \\cdot 600 = 0 $$","blockMath-rFisvPiQJ",{"__TSPROSE_proseElement":175,"schema":6013,"children":6014,"id":6031},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6015,6018,6022,6025,6029],{"__TSPROSE_proseElement":175,"schema":6016,"data":6017},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"We got a quadratic equation. Since coefficient ",{"__TSPROSE_proseElement":175,"schema":6019,"data":6020,"storageKey":1341,"id":6021},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1340},"inlinerMath-RvASDBkSH-2",{"__TSPROSE_proseElement":175,"schema":6023,"data":6024},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is even, solve it using the ",{"__TSPROSE_proseElement":175,"schema":6026,"data":6027,"storageKey":1350,"id":6028},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":1349},"dependencyInliner-pF48j3mqL-2",{"__TSPROSE_proseElement":175,"schema":6030,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-AcD7i4d3q",{"__TSPROSE_proseElement":175,"schema":6033,"data":6034,"storageKey":6036,"id":6037},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6035,"freeze":246},"M = -300 \\pm \\sqrt{300^2 + 450 \\cdot 600} = -300 \\pm 600","$$ M = -300 \\pm \\sqrt{300^2 + 450 \\cdot 600} = -300 \\pm 600 $$","blockMath-p1O0bf2wP",{"__TSPROSE_proseElement":175,"schema":6039,"children":6040,"id":6067},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6041,6043,6049,6051,6057,6060,6064],{"__TSPROSE_proseElement":175,"schema":6042,"data":2593},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":6044,"data":6045,"storageKey":6047,"id":6048},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6046},"M = 300","$ M = 300 $","inlinerMath-kqLRiqfNk",{"__TSPROSE_proseElement":175,"schema":6050,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":6052,"data":6053,"storageKey":6055,"id":6056},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6054},"M = -900","$ M = -900 $","inlinerMath-ZnO2r7U0j",{"__TSPROSE_proseElement":175,"schema":6058,"data":6059},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Negative mass is nonsense, so ",{"__TSPROSE_proseElement":175,"schema":6061,"data":6062,"storageKey":6047,"id":6063},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6046},"inlinerMath-kqLRiqfNk-1",{"__TSPROSE_proseElement":175,"schema":6065,"data":6066},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," grams. Then the original concentration is:","paragraph-LFc8RJkav",{"__TSPROSE_proseElement":175,"schema":6069,"data":6070,"storageKey":6072,"id":6073},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6071,"freeze":246},"c = \\frac{18}{300} = \\frac{6}{100} = 0.06","$$ c = \\frac{18}{300} = \\frac{6}{100} = 0.06 $$","blockMath-IUzJ8GXtM","salt-solution",{"__TSPROSE_proseElement":175,"schema":6076,"data":6077,"children":6081,"id":6398},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":6078},{"title":6079,"level":1141,"attributes":6080},"Metal Ingot",[],[6082,6124,6136,6146,6182,6199],{"__TSPROSE_proseElement":175,"schema":6083,"children":6084},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[6085],{"__TSPROSE_proseElement":175,"schema":6086,"children":6087,"id":6123},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6088,6091,6097,6100,6104,6107,6111,6114,6120],{"__TSPROSE_proseElement":175,"schema":6089,"data":6090},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"There are two copper alloys with another metal, and the relative copper content in one of them is ",{"__TSPROSE_proseElement":175,"schema":6092,"data":6093,"storageKey":6095,"id":6096},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6094},"40%","$ 40% $","inlinerMath-KOj3WSN52",{"__TSPROSE_proseElement":175,"schema":6098,"data":6099},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," lower than in the other. A piece of the first alloy containing ",{"__TSPROSE_proseElement":175,"schema":6101,"data":6102,"storageKey":619,"id":6103},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":618},"inlinerMath-9rSWhNw25-12",{"__TSPROSE_proseElement":175,"schema":6105,"data":6106},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," kg of copper is melted together with a piece of the second alloy containing ",{"__TSPROSE_proseElement":175,"schema":6108,"data":6109,"storageKey":889,"id":6110},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":888},"inlinerMath-iwXZWeXrQ-3",{"__TSPROSE_proseElement":175,"schema":6112,"data":6113},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," kg of copper. The resulting ingot contains ",{"__TSPROSE_proseElement":175,"schema":6115,"data":6116,"storageKey":6118,"id":6119},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6117},"36%","$ 36% $","inlinerMath-BJSWTlkYd",{"__TSPROSE_proseElement":175,"schema":6121,"data":6122},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," copper. Find the copper percentage in the first alloy.","paragraph-CcxNIgjdk",{"__TSPROSE_proseElement":175,"schema":6125,"data":6126},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":6127,"serializedValidator":6128},"Copper percentage in first alloy",{"type":5884,"answer":6129},[6130,6133],{"__ERUDIT_CHECK":175,"name":396,"data":6131},{"expr":6132},"0.2",{"__ERUDIT_CHECK":175,"name":396,"data":6134},{"expr":6135},"20%",{"__TSPROSE_proseElement":175,"schema":6137,"children":6138},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[6139],{"__TSPROSE_proseElement":175,"schema":6140,"children":6141,"id":6145},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6142],{"__TSPROSE_proseElement":175,"schema":6143,"data":6144},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"First find the total mass of the resulting ingot.","paragraph-z9qiXV2Wd",{"__TSPROSE_proseElement":175,"schema":6147,"children":6148},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[6149],{"__TSPROSE_proseElement":175,"schema":6150,"children":6151,"id":6181},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6152,6155,6161,6164,6170,6172,6178],{"__TSPROSE_proseElement":175,"schema":6153,"data":6154},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The mass of the ingot is the sum of the masses of the two pieces, ",{"__TSPROSE_proseElement":175,"schema":6156,"data":6157,"storageKey":6159,"id":6160},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6158},"m_1 + m_2 = M","$ m_1 + m_2 = M $","inlinerMath-fB8YX9Uq1",{"__TSPROSE_proseElement":175,"schema":6162,"data":6163},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Replace ",{"__TSPROSE_proseElement":175,"schema":6165,"data":6166,"storageKey":6168,"id":6169},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6167},"m_1","$ m_1 $","inlinerMath-G8HJosHhe",{"__TSPROSE_proseElement":175,"schema":6171,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":6173,"data":6174,"storageKey":6176,"id":6177},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6175},"m_2","$ m_2 $","inlinerMath-3pWm1fGIz",{"__TSPROSE_proseElement":175,"schema":6179,"data":6180},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," with expressions through the copper concentrations of the two alloys.","paragraph-Rpg0J7M0B",{"__TSPROSE_proseElement":175,"schema":6183,"children":6184},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[6185],{"__TSPROSE_proseElement":175,"schema":6186,"children":6187,"id":6198},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6188,6191,6196],{"__TSPROSE_proseElement":175,"schema":6189,"data":6190},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"The copper percentage in the first alloy is ",{"__TSPROSE_proseElement":175,"schema":6192,"data":6193,"storageKey":6194,"id":6195},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6135},"$ 20% $","inlinerMath-uSJTYChps",{"__TSPROSE_proseElement":175,"schema":6197,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-o4PPS4zRo",{"__TSPROSE_proseElement":175,"schema":6200,"children":6201},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[6202,6216,6222,6229,6235,6242,6248,6255,6261,6268,6274,6281,6287,6294,6300,6312,6318,6331,6337],{"__TSPROSE_proseElement":175,"schema":6203,"children":6204,"id":6215},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6205,6208,6212],{"__TSPROSE_proseElement":175,"schema":6206,"data":6207},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Find the total mass ",{"__TSPROSE_proseElement":175,"schema":6209,"data":6210,"storageKey":5939,"id":6211},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":5938},"inlinerMath-0MhrSCdD8-1",{"__TSPROSE_proseElement":175,"schema":6213,"data":6214},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," of the resulting ingot. You can get it from the concentration formula for the ingot:","paragraph-WahNM6X3Q",{"__TSPROSE_proseElement":175,"schema":6217,"data":6218,"storageKey":6220,"id":6221},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6219,"freeze":246},"0.36 = \\frac{6 + 12}{M} \\implies M = \\frac{18}{0.36} = \\frac{18 \\cdot 100}{36} = 50","$$ 0.36 = \\frac{6 + 12}{M} \\implies M = \\frac{18}{0.36} = \\frac{18 \\cdot 100}{36} = 50 $$","blockMath-7BaMXVaEn",{"__TSPROSE_proseElement":175,"schema":6223,"children":6224,"id":6228},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6225],{"__TSPROSE_proseElement":175,"schema":6226,"data":6227},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Write the relation between the original concentrations:","paragraph-K6972iPlW",{"__TSPROSE_proseElement":175,"schema":6230,"data":6231,"storageKey":6233,"id":6234},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6232,"freeze":246},"c_1 = c_2 - 0.4","$$ c_1 = c_2 - 0.4 $$","blockMath-6WDVoyNdw",{"__TSPROSE_proseElement":175,"schema":6236,"children":6237,"id":6241},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6238],{"__TSPROSE_proseElement":175,"schema":6239,"data":6240},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"That alone does not help much yet, because concentration is exactly what we are trying to find. So switch to a different route. The ingot came from two alloy pieces, so:","paragraph-D7UmnBe6F",{"__TSPROSE_proseElement":175,"schema":6243,"data":6244,"storageKey":6246,"id":6247},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6245,"freeze":246},"m_1 + m_2 = 50","$$ m_1 + m_2 = 50 $$","blockMath-3rUQ9jK6J",{"__TSPROSE_proseElement":175,"schema":6249,"children":6250,"id":6254},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6251],{"__TSPROSE_proseElement":175,"schema":6252,"data":6253},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"But the mass of each piece is directly tied to the copper concentration in that piece:","paragraph-E61d6YnVA",{"__TSPROSE_proseElement":175,"schema":6256,"data":6257,"storageKey":6259,"id":6260},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6258,"freeze":246},"c_1 = \\frac{6}{m_1} \\implies m_1 = \\frac{6}{c_1} >>{big} c_2 = \\frac{12}{m_2} \\implies m_2 = \\frac{12}{c_2}","$$ c_1 = \\frac{6}{m_1} \\implies m_1 = \\frac{6}{c_1} >>{big} c_2 = \\frac{12}{m_2} \\implies m_2 = \\frac{12}{c_2} $$","blockMath-dX20hBUQv",{"__TSPROSE_proseElement":175,"schema":6262,"children":6263,"id":6267},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6264],{"__TSPROSE_proseElement":175,"schema":6265,"data":6266},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Substitute these fractions for the masses:","paragraph-7fgs3f2AH",{"__TSPROSE_proseElement":175,"schema":6269,"data":6270,"storageKey":6272,"id":6273},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6271,"freeze":246},"\\frac{6}{c_1} + \\frac{12}{c_2} = 50","$$ \\frac{6}{c_1} + \\frac{12}{c_2} = 50 $$","blockMath-Qs5jyMxjm",{"__TSPROSE_proseElement":175,"schema":6275,"children":6276,"id":6280},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6277],{"__TSPROSE_proseElement":175,"schema":6278,"data":6279},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Now replace the first concentration by its expression through the second:","paragraph-CMZtY293E",{"__TSPROSE_proseElement":175,"schema":6282,"data":6283,"storageKey":6285,"id":6286},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6284,"freeze":246},"\\frac{6}{c_2 - 0.4} + \\frac{12}{c_2} = 50","$$ \\frac{6}{c_2 - 0.4} + \\frac{12}{c_2} = 50 $$","blockMath-eQZPiQDRQ",{"__TSPROSE_proseElement":175,"schema":6288,"children":6289,"id":6293},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6290],{"__TSPROSE_proseElement":175,"schema":6291,"data":6292},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Great. Now it is a one-variable equation. Solve it:","paragraph-pZa6LrWIV",{"__TSPROSE_proseElement":175,"schema":6295,"data":6296,"storageKey":6298,"id":6299},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6297,"freeze":246},"\\frac{6}{c_2 - \\frac{4}{10}} + \\frac{12}{c_2} = 50 \\\\ \\frac{3}{c_2 - \\frac{4}{10}} + \\frac{6}{c_2} = 25 \\\\ 3c_2 + 6\\left(c_2 - \\frac{4}{10}\\right) = 25c_2\\left(c_2 - \\frac{4}{10}\\right) \\\\ 3c_2 + 6c_2 - \\frac{24}{10} = 25c_2^2 - 10c_2 \\\\ 0 = 25c_2^2 - 19c_2 + \\frac{12}{5}","$$ \\frac{6}{c_2 - \\frac{4}{10}} + \\frac{12}{c_2} = 50 \\\\ \\frac{3}{c_2 - \\frac{4}{10}} + \\frac{6}{c_2} = 25 \\\\ 3c_2 + 6\\left(c_2 - \\frac{4}{10}\\right) = 25c_2\\left(c_2 - \\frac{4}{10}\\right) \\\\ 3c_2 + 6c_2 - \\frac{24}{10} = 25c_2^2 - 10c_2 \\\\ 0 = 25c_2^2 - 19c_2 + \\frac{12}{5} $$","blockMath-VhSeH38U5",{"__TSPROSE_proseElement":175,"schema":6301,"children":6302,"id":6311},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6303,6305,6309],{"__TSPROSE_proseElement":175,"schema":6304,"data":3592},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":6306,"data":6307,"storageKey":2427,"id":6308},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":193},"dependencyInliner-emFvGyx3M-5",{"__TSPROSE_proseElement":175,"schema":6310,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-AirhIkDv2-1",{"__TSPROSE_proseElement":175,"schema":6313,"data":6314,"storageKey":6316,"id":6317},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6315,"freeze":246},"D = (-19)^2 - 4 \\cdot 25 \\cdot \\frac{12}{5} = 361 - 240 = 121","$$ D = (-19)^2 - 4 \\cdot 25 \\cdot \\frac{12}{5} = 361 - 240 = 121 $$","blockMath-B7oYzteU1",{"__TSPROSE_proseElement":175,"schema":6319,"children":6320,"id":6330},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6321,6324,6328],{"__TSPROSE_proseElement":175,"schema":6322,"data":6323},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"And the roots via the ",{"__TSPROSE_proseElement":175,"schema":6325,"data":6326,"storageKey":2448,"id":6327},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":3258},"dependencyInliner-1X2UJbegR-3",{"__TSPROSE_proseElement":175,"schema":6329,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-EoxZ8m7fq",{"__TSPROSE_proseElement":175,"schema":6332,"data":6333,"storageKey":6335,"id":6336},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6334,"freeze":246},"c_2 = \\frac{-(-19) \\pm \\sqrt{121}}{2 \\cdot 25} = \\frac{19 \\pm 11}{50}","$$ c_2 = \\frac{-(-19) \\pm \\sqrt{121}}{2 \\cdot 25} = \\frac{19 \\pm 11}{50} $$","blockMath-QOpJOwP7M",{"__TSPROSE_proseElement":175,"schema":6338,"children":6339,"id":6397},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6340,6342,6348,6350,6356,6359,6363,6366,6372,6375,6379,6382,6388,6391,6395],{"__TSPROSE_proseElement":175,"schema":6341,"data":2593},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":6343,"data":6344,"storageKey":6346,"id":6347},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6345},"c_2 = 0.6","$ c_2 = 0.6 $","inlinerMath-Uy914ip7y",{"__TSPROSE_proseElement":175,"schema":6349,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":6351,"data":6352,"storageKey":6354,"id":6355},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6353},"c_2 = 0.16","$ c_2 = 0.16 $","inlinerMath-vHwshtYGl",{"__TSPROSE_proseElement":175,"schema":6357,"data":6358},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". If ",{"__TSPROSE_proseElement":175,"schema":6360,"data":6361,"storageKey":6354,"id":6362},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6353},"inlinerMath-vHwshtYGl-1",{"__TSPROSE_proseElement":175,"schema":6364,"data":6365},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", then ",{"__TSPROSE_proseElement":175,"schema":6367,"data":6368,"storageKey":6370,"id":6371},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6369},"c_1 = -0.24","$ c_1 = -0.24 $","inlinerMath-isnd04NJA",{"__TSPROSE_proseElement":175,"schema":6373,"data":6374},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", which is impossible because concentration cannot be negative. So the only valid choice is ",{"__TSPROSE_proseElement":175,"schema":6376,"data":6377,"storageKey":6346,"id":6378},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6345},"inlinerMath-Uy914ip7y-1",{"__TSPROSE_proseElement":175,"schema":6380,"data":6381},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},", and hence ",{"__TSPROSE_proseElement":175,"schema":6383,"data":6384,"storageKey":6386,"id":6387},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6385},"c_1 = 0.2","$ c_1 = 0.2 $","inlinerMath-PfqS7fQmh",{"__TSPROSE_proseElement":175,"schema":6389,"data":6390},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Therefore the copper percentage in the first alloy is ",{"__TSPROSE_proseElement":175,"schema":6392,"data":6393,"storageKey":6194,"id":6394},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6135},"inlinerMath-uSJTYChps-1",{"__TSPROSE_proseElement":175,"schema":6396,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-n3MIH3c2x","metal-ingot",{"__TSPROSE_proseElement":175,"schema":6400,"data":6401,"children":6405,"id":6647},{"name":337,"type":245,"linkable":175,"__TSPROSE_schema":175},{"info":6402},{"title":6403,"level":1419,"attributes":6404},"Gas Mixtures",[],[6406,6441,6447,6457,6467,6484],{"__TSPROSE_proseElement":175,"schema":6407,"children":6408},{"name":346,"type":245,"linkable":246,"__TSPROSE_schema":175},[6409],{"__TSPROSE_proseElement":175,"schema":6410,"children":6411,"id":6440},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6412,6415,6419,6422,6428,6431,6437],{"__TSPROSE_proseElement":175,"schema":6413,"data":6414},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"A vessel with capacity ",{"__TSPROSE_proseElement":175,"schema":6416,"data":6417,"storageKey":812,"id":6418},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":811},"inlinerMath-fBsW71uIl-6",{"__TSPROSE_proseElement":175,"schema":6420,"data":6421},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," liters is filled with air containing ",{"__TSPROSE_proseElement":175,"schema":6423,"data":6424,"storageKey":6426,"id":6427},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6425},"16%","$ 16% $","inlinerMath-hJ2sVV4MT",{"__TSPROSE_proseElement":175,"schema":6429,"data":6430},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," oxygen. Some amount of air is let out and the same amount of nitrogen is pumped in. Then the same amount of the new mixture is let out again, and again replaced by the same amount of nitrogen. In the final mixture, the oxygen concentration is ",{"__TSPROSE_proseElement":175,"schema":6432,"data":6433,"storageKey":6435,"id":6436},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6434},"9%","$ 9% $","inlinerMath-zzWqNvagf",{"__TSPROSE_proseElement":175,"schema":6438,"data":6439},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". How many liters are released each time?","paragraph-aEb1a6JBc",{"__TSPROSE_proseElement":175,"schema":6442,"data":6443},{"name":392,"type":245,"linkable":246,"__TSPROSE_schema":175},{"label":6444,"serializedValidator":6445},"Liters released each time",{"__ERUDIT_CHECK":175,"name":396,"data":6446},{"expr":398},{"__TSPROSE_proseElement":175,"schema":6448,"children":6449},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[6450],{"__TSPROSE_proseElement":175,"schema":6451,"children":6452,"id":6456},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6453],{"__TSPROSE_proseElement":175,"schema":6454,"data":6455},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Find the expression for the oxygen concentration after the first stage.","paragraph-lvYSbvA0H",{"__TSPROSE_proseElement":175,"schema":6458,"children":6459},{"name":401,"type":245,"linkable":246,"__TSPROSE_schema":175},[6460],{"__TSPROSE_proseElement":175,"schema":6461,"children":6462,"id":6466},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6463],{"__TSPROSE_proseElement":175,"schema":6464,"data":6465},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Use that first-stage expression to build the oxygen concentration after the second stage.","paragraph-z3yWPlOOn",{"__TSPROSE_proseElement":175,"schema":6468,"children":6469},{"name":412,"type":245,"linkable":246,"__TSPROSE_schema":175},[6470],{"__TSPROSE_proseElement":175,"schema":6471,"children":6472,"id":6483},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6473,6476,6480],{"__TSPROSE_proseElement":175,"schema":6474,"data":6475},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Each time, ",{"__TSPROSE_proseElement":175,"schema":6477,"data":6478,"storageKey":423,"id":6479},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":398},"inlinerMath-z9Yar9waf-22",{"__TSPROSE_proseElement":175,"schema":6481,"data":6482},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," liters of mixture are released.","paragraph-zTWgxpEmn",{"__TSPROSE_proseElement":175,"schema":6485,"children":6486},{"name":431,"type":245,"linkable":246,"__TSPROSE_schema":175},[6487,6519,6525,6572,6578,6593,6599],{"__TSPROSE_proseElement":175,"schema":6488,"children":6489,"id":6518},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6490,6493,6497,6500,6506,6509,6515],{"__TSPROSE_proseElement":175,"schema":6491,"data":6492},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"At the first stage, ",{"__TSPROSE_proseElement":175,"schema":6494,"data":6495,"storageKey":443,"id":6496},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":442},"inlinerMath-TLSwLL8iH-9",{"__TSPROSE_proseElement":175,"schema":6498,"data":6499},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," liters of air are released and replaced with nitrogen. The vessel still contains ",{"__TSPROSE_proseElement":175,"schema":6501,"data":6502,"storageKey":6504,"id":6505},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6503},"8-x","$ 8-x $","inlinerMath-qctTAozbk",{"__TSPROSE_proseElement":175,"schema":6507,"data":6508},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," liters of the old mixture, and therefore ",{"__TSPROSE_proseElement":175,"schema":6510,"data":6511,"storageKey":6513,"id":6514},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6512},"(8-x) \\cdot 0.16","$ (8-x) \\cdot 0.16 $","inlinerMath-Ayvn6EnFa",{"__TSPROSE_proseElement":175,"schema":6516,"data":6517},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," liters of oxygen. No new oxygen was added. So the oxygen concentration after the first stage is:","paragraph-x1E2tZ2Ml",{"__TSPROSE_proseElement":175,"schema":6520,"data":6521,"storageKey":6523,"id":6524},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6522,"freeze":246},"\\frac{(8-x) \\cdot 0.16}{8} = (8-x) \\cdot 0.02","$$ \\frac{(8-x) \\cdot 0.16}{8} = (8-x) \\cdot 0.02 $$","blockMath-hZ6yB1wyn",{"__TSPROSE_proseElement":175,"schema":6526,"children":6527,"id":6571},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6528,6531,6535,6538,6542,6545,6551,6554,6560,6563,6569],{"__TSPROSE_proseElement":175,"schema":6529,"data":6530},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"At the second stage, ",{"__TSPROSE_proseElement":175,"schema":6532,"data":6533,"storageKey":443,"id":6534},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":442},"inlinerMath-TLSwLL8iH-10",{"__TSPROSE_proseElement":175,"schema":6536,"data":6537},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," liters are released again and replaced with nitrogen. The vessel now contains ",{"__TSPROSE_proseElement":175,"schema":6539,"data":6540,"storageKey":6504,"id":6541},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6503},"inlinerMath-qctTAozbk-1",{"__TSPROSE_proseElement":175,"schema":6543,"data":6544},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," liters of the first-stage mixture, whose oxygen concentration is ",{"__TSPROSE_proseElement":175,"schema":6546,"data":6547,"storageKey":6549,"id":6550},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6548},"0.16 - 0.02x","$ 0.16 - 0.02x $","inlinerMath-L7kkVTUkv",{"__TSPROSE_proseElement":175,"schema":6552,"data":6553},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". So the volume of oxygen at the second stage is ",{"__TSPROSE_proseElement":175,"schema":6555,"data":6556,"storageKey":6558,"id":6559},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6557},"(8-x) \\cdot (8-x) \\cdot 0.02","$ (8-x) \\cdot (8-x) \\cdot 0.02 $","inlinerMath-7tKs7Tcu3",{"__TSPROSE_proseElement":175,"schema":6561,"data":6562},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". Divide by the total volume and set it to the required concentration ",{"__TSPROSE_proseElement":175,"schema":6564,"data":6565,"storageKey":6567,"id":6568},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6566},"0.09","$ 0.09 $","inlinerMath-4irZg4DDi",{"__TSPROSE_proseElement":175,"schema":6570,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-syxT0mJbA",{"__TSPROSE_proseElement":175,"schema":6573,"data":6574,"storageKey":6576,"id":6577},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6575,"freeze":246},"\\frac{(8-x) \\cdot (8-x) \\cdot 0.02}{8} = 0.09 \\\\ \\frac{(8-x)^2 \\cdot 0.02}{8} = 0.09 \\\\","$$ \\frac{(8-x) \\cdot (8-x) \\cdot 0.02}{8} = 0.09 \\\\ \\frac{(8-x)^2 \\cdot 0.02}{8} = 0.09 \\\\ $$","blockMath-9bEWJ7kKP",{"__TSPROSE_proseElement":175,"schema":6579,"children":6580,"id":6592},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6581,6583,6589],{"__TSPROSE_proseElement":175,"schema":6582,"data":1990},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":6584,"data":6585,"storageKey":6587,"id":6588},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6586},"100","$ 100 $","inlinerMath-EMZ1ocF1g",{"__TSPROSE_proseElement":175,"schema":6590,"data":6591},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," to remove decimals:","paragraph-U0DluONPm",{"__TSPROSE_proseElement":175,"schema":6594,"data":6595,"storageKey":6597,"id":6598},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":6596,"freeze":246},"\\frac{(8-x)^2 \\cdot 2}{8} = 9 \\\\ (8-x)^2 = 36 \\\\ 8-x = \\pm 6 \\\\ 8\\mp 6 = x","$$ \\frac{(8-x)^2 \\cdot 2}{8} = 9 \\\\ (8-x)^2 = 36 \\\\ 8-x = \\pm 6 \\\\ 8\\mp 6 = x $$","blockMath-oe1biUp3i",{"__TSPROSE_proseElement":175,"schema":6600,"children":6601,"id":6646},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[6602,6604,6608,6610,6616,6619,6623,6626,6630,6633,6637,6640,6644],{"__TSPROSE_proseElement":175,"schema":6603,"data":2593},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":6605,"data":6606,"storageKey":1123,"id":6607},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1122},"inlinerMath-485ZM4Vn2-1",{"__TSPROSE_proseElement":175,"schema":6609,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":6611,"data":6612,"storageKey":6614,"id":6615},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":6613},"x = 14","$ x = 14 $","inlinerMath-Dr00vniUT",{"__TSPROSE_proseElement":175,"schema":6617,"data":6618},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},". 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2)^2 - (b - 2)^2 = 28","$$ (a - 2)^2 - (b - 2)^2 = 28 $$","blockMath-M6jdi4RJS",{"__TSPROSE_proseElement":175,"schema":8461,"children":8462,"id":8466},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[8463],{"__TSPROSE_proseElement":175,"schema":8464,"data":8465},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"Use the difference of squares formula:","paragraph-I6V7a4zVm",{"__TSPROSE_proseElement":175,"schema":8468,"data":8469,"storageKey":8471,"id":8472},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":8470,"freeze":246},"(a - 2 - (b - 2))(a - 2 + (b - 2)) = 28 \\\\ (a - b)(a + b - 4) = 28","$$ (a - 2 - (b - 2))(a - 2 + (b - 2)) = 28 \\\\ (a - b)(a + b - 4) = 28 $$","blockMath-4EFtCmtNM",{"__TSPROSE_proseElement":175,"schema":8474,"children":8475,"id":8493},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[8476,8478,8484,8486,8490],{"__TSPROSE_proseElement":175,"schema":8477,"data":6743},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":8479,"data":8480,"storageKey":8482,"id":8483},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":8481},"\\frac{5}{4}b","$ \\frac{5}{4}b $","inlinerMath-UkVbUK3Pq",{"__TSPROSE_proseElement":175,"schema":8485,"data":6750},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":8487,"data":8488,"storageKey":1511,"id":8489},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1510},"inlinerMath-YqDfaaRXA-14",{"__TSPROSE_proseElement":175,"schema":8491,"data":8492},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," and simplify:","paragraph-h1oI57ks7",{"__TSPROSE_proseElement":175,"schema":8495,"data":8496,"storageKey":8498,"id":8499},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":8497,"freeze":246},"\\left(\\frac{5}{4}b - b\\right)\\left(\\frac{5}{4}b + b - 4\\right) = 28 \\\\ \\left(\\frac{1}{4}b\\right)\\left(\\frac{9}{4}b - 4\\right) = 28 \\\\ \\frac{9}{16}b^2 - b = 28 \\\\ 9b^2 - 16b = 448 \\\\ 9b^2 - 16b - 448 = 0","$$ \\left(\\frac{5}{4}b - b\\right)\\left(\\frac{5}{4}b + b - 4\\right) = 28 \\\\ \\left(\\frac{1}{4}b\\right)\\left(\\frac{9}{4}b - 4\\right) = 28 \\\\ \\frac{9}{16}b^2 - b = 28 \\\\ 9b^2 - 16b = 448 \\\\ 9b^2 - 16b - 448 = 0 $$","blockMath-oktbmc5pw",{"__TSPROSE_proseElement":175,"schema":8501,"children":8502,"id":8518},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[8503,8505,8509,8512,8516],{"__TSPROSE_proseElement":175,"schema":8504,"data":6017},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":8506,"data":8507,"storageKey":1341,"id":8508},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":1340},"inlinerMath-RvASDBkSH-3",{"__TSPROSE_proseElement":175,"schema":8510,"data":8511},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," is even, use the ",{"__TSPROSE_proseElement":175,"schema":8513,"data":8514,"storageKey":1350,"id":8515},{"name":493,"type":255,"linkable":175,"__TSPROSE_schema":175},{"label":1349},"dependencyInliner-pF48j3mqL-3",{"__TSPROSE_proseElement":175,"schema":8517,"data":1039},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-57yM7RsGT",{"__TSPROSE_proseElement":175,"schema":8520,"data":8521,"storageKey":8523,"id":8524},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":8522,"freeze":246},"b = \\frac{-(-8) \\pm \\sqrt{(-8)^2 - 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It may look as if the two sides are separately proportional to ",{"__TSPROSE_proseElement":175,"schema":8588,"data":8589,"storageKey":796,"id":8590},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":795},"inlinerMath-MAD3UaxRg-12",{"__TSPROSE_proseElement":175,"schema":8592,"data":527},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},{"__TSPROSE_proseElement":175,"schema":8594,"data":8595,"storageKey":610,"id":8596},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":609},"inlinerMath-ZXyaHjumL-12",{"__TSPROSE_proseElement":175,"schema":8598,"data":8599},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175}," with different coefficients.","paragraph-lBSsLPCaJ",{"__TSPROSE_proseElement":175,"schema":8602,"data":8603,"storageKey":8605,"id":8606},{"name":458,"type":245,"linkable":175,"__TSPROSE_schema":175},{"katex":8604,"freeze":246},"a = 5x >>{big} b = 4y >>{big} x \\neq y","$$ a = 5x >>{big} b = 4y >>{big} x \\neq y $$","blockMath-FRGonTwi1",{"__TSPROSE_proseElement":175,"schema":8608,"children":8609,"id":8621},{"name":250,"type":245,"linkable":175,"__TSPROSE_schema":175},[8610,8613,8619],{"__TSPROSE_proseElement":175,"schema":8611,"data":8612},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"But the statement should be read literally: you are supposed to form the ratio ",{"__TSPROSE_proseElement":175,"schema":8614,"data":8615,"storageKey":8617,"id":8618},{"name":356,"type":255,"linkable":175,"__TSPROSE_schema":175},{"katex":8616},"a : b = 5 : 4","$ a : b = 5 : 4 $","inlinerMath-sUhPmojzI",{"__TSPROSE_proseElement":175,"schema":8620,"data":536},{"name":254,"type":255,"linkable":246,"__TSPROSE_schema":175},"paragraph-ioB7y0nhw","mysterious-squares",{"$ 40 $":8624,"$ 30 $":8624,"$ 936 $":8624,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fbox-sketch.svg":8625,"$ 2 $":8624,"$ x $":8624,"$$ 40 - x - x = 40 - 2x $$":8624,"$ 30 - 2x $":8624,"$$ (40 - 2x)(30 - 2x) = 936 \\\\ (20 - x)(30 - 2x) = 468 \\\\ (20 - x)(15 - x) = 234 \\\\ 300 - 35x + x^2 = 234 \\\\ x^2 - 35x + 66 = 0 $$":8624,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Ffactoring\u002Farticle\u002F$manualFactoring":8629,"$ -35 $":8624,"$ 66 $":8624,"$ -33 $":8624,"$ -2 $":8624,"$$ (x - 33)(x - 2) = 0 $$":8624,"$ 33 $":8624,"$ 120 $":8624,"$ 4 $":8624,"$ 6 $":8624,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fsteel-frame.svg":8635,"$ 3 $":8624,"$ 4 + 2x $":8624,"$ 6 + 2x $":8624,"$$ S = (4 + 2x)(6 + 2x) $$":8624,"$$ 120 = (4 + 2x)(6 + 2x) $$":8624,"$$ 120 = 24 + 8x + 12x + 4x^2 \\\\ 120 = 24 + 20x + 4x^2 $$":8624,"$$ 120 - 120 = 24 + 20x + 4x^2 - 120 \\\\ 0 = 4x^2 + 20x - 96 $$":8624,"$$ 0 = x^2 + 5x - 24 $$":8624,"$ 5 $":8624,"$ -24 $":8624,"$ 8 $":8624,"$ -3 $":8624,"$$ (x + 8)(x - 3) = 0 $$":8624,"$ -8 $":8624,"$ 12 $":8624,"$ 2x $":8624,"$ (8 + 2x) $":8624,"$ (12 + 2x) $":8624,"$ 8 \\cdot 12 = 96 $":8624,"$ 2 \\cdot 96 = 192 $":8624,"$$ (8 + 2x)(12 + 2x) = 192 $$":8624,"$$ 96 + 16x + 24x + 4x^2 = 192 \\\\ 4x^2 + 40x + 96 = 192 \\\\ 4x^2 + 40x - 96 = 0 $$":8624,"$$ x^2 + 10x - 24 = 0 $$":8624,"$ C $":8624,"$ 10 $":8624,"$$ (x + 12)(x - 2) = 0 $$":8624,"$ x = -12 $":8624,"$ x = 2 $":8624,"$ 840 $":8624,"$ 20 $":8624,"$ N_1 $":8624,"$ N_2 = N_1 + 1 $":8624,"$ 7 $":8624,"$ N_2 $":8624,"$$ N_2 = N_1 + 1 $$":8624,"$ P $":8624,"$ P - 20 $":8624,"$$ \\frac{840}{P - 20} = \\frac{840}{P} + 1 $$":8624,"$ P(P - 20) $":8624,"$$ 840P = 840(P - 20) + P(P - 20) $$":8624,"$$ \\cancel{840P} = \\cancel{840P} - 20 \\cdot 840 + P^2 - 20P \\\\ 0 = - 20 \\cdot 840 + P^2 - 20P \\\\ 0 = P^2 - 20P - 20 \\cdot 840 $$":8624,"$ B $":8624,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002Fpractice\u002F$evenB":8639,"$$ P = -(-10) \\pm \\sqrt{(-10)^2 - (-20\\cdot 840)} = 10 \\pm \\sqrt{100 + 2 \\cdot 84 \\cdot 100} = 10 \\pm \\sqrt{100(1 + 168)} = 10 \\pm \\sqrt{100}\\sqrt{169} = 10 \\pm 130 $$":8624,"$ P = 140 $":8624,"$ P = -120 $":8624,"$$ N_1 = \\frac{840}{140} = 6 $$":8624,"$ N_2 = 7 $":8624,"$ 56 $":8624,"$ 11 $":8624,"$ p $":8624,"$ a $":8624,"$$ p-6 = 2(a-6)^2 $$":8624,"$$ a + 4 = \\frac{1}{4}(p + 4) \\\\ 4a + 16 = p + 4 $$":8624,"$ p - 6 $":8624,"$$ 4a + 16 - 10 = p + 4 - 10 \\\\ 4a + 6 = p - 6 $$":8624,"$ 4a + 6 $":8624,"$$ 4a + 6 = 2(a - 6)^2 \\\\ 4a + 6 = 2(a^2 - 12a + 36) \\\\ 4a + 6 = 2a^2 - 24a + 72 \\\\ 0 = 2a^2 - 28a + 66 \\\\ $$":8624,"$$ 0 = a^2 - 14a + 33 $$":8624,"$ A $":8624,"$ 1 $":8624,"$ -11 $":8624,"$ -14 $":8624,"$$ (a - 11)(a - 3) = 0 $$":8624,"$ 4a + 6 = p - 6 $":8624,"$$ 4 \\cdot 11 + 6 = p - 6 \\\\ 44 + 6 = p - 6 \\\\ 50 = p - 6 \\\\ p = 56 $$":8624,"$ 15 $":8624,"$ 36 $":8624,"$ y $":8624,"$ xy = 36 $":8624,"$$ \\frac{15}{x-1} >>{big} \\frac{10}{y-1} $$":8624,"$$ \\frac{15}{x-1} = \\frac{10}{y-1} + 1 $$":8624,"$ y = \\frac{36}{x} $":8624,"$$ \\frac{15}{x-1} = \\frac{10}{\\frac{36}{x}-1} + 1 \\\\ \\frac{15}{x-1} = \\frac{10}{\\frac{36-x}{x}} + 1 \\\\ \\frac{15}{x-1} = \\frac{10x}{36-x} + 1 $$":8624,"$ (x-1)(36-x) $":8624,"$$ 15(36-x) = 10x(x-1) + (x-1)(36-x) $$":8624,"$$ 540 - 15x = 10x^2 - 10x + 36x - x^2 - 36 + x \\\\ 540 - 15x = 9x^2 + 27x - 36 \\\\ 0 = 9x^2 + 42x - 576 $$":8624,"$$ 3x^2 + 14x - 192 = 0 $$":8624,"$ B = 14 $":8624,"$$ x = \\frac{-7 \\pm \\sqrt{7^2 - 3 \\cdot (-192)}}{3} = \\frac{-7 \\pm \\sqrt{49 + 576}}{3} = \\frac{-7 \\pm \\sqrt{625}}{3} = \\frac{-7 \\pm 25}{3} $$":8624,"$ x = 6 $":8624,"$ x = -\\frac{32}{3} $":8624,"$$ y = \\frac{36}{x} = \\frac{36}{6} = 6 $$":8624,"$$ \\frac{15}{6-1} = \\frac{15}{5} = 3 >>{big} \\frac{10}{6-1} = \\frac{10}{5} = 2 $$":8624,"$ g = 10 $":8624,"$ h_0 $":8624,"$ v_0 $":8624,"$ h $":8624,"$ t $":8624,"$$ h(t) = h_0 + v_0 t - \\frac{g}{2}t^2 $$":8624,"$ 60 $":8624,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fpisa-experiment.svg":8644,"$$ h(t) = 60 - 5t - \\frac{10}{2}t^2 = 60 - 5t - 5t^2 $$":8624,"$$ 60 - 5t - 5t^2 = 0 $$":8624,"$ -5 $":8624,"$ t^2 $":8624,"$$ t^2 + t - 12 = 0 $$":8624,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002Farticle\u002F$discriminant":8648,"$$ D = 1^2 - 4 \\cdot 1 \\cdot (-12) = 1 + 48 = 49 $$":8624,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002Farticle\u002F$formula":8651,"$$ t = \\frac{-1 \\pm \\sqrt{49}}{2 \\cdot 1} = \\frac{-1 \\pm 7}{2} $$":8624,"$ t = -4 $":8624,"$ t = 3 $":8624,"$$ h(t) = 60 + 5t - \\frac{10}{2}t^2 = 60 + 5t - 5t^2 $$":8624,"$$ 60 + 5t - 5t^2 = 0 $$":8624,"$$ t^2 - t - 12 = 0 $$":8624,"$ -1 $":8624,"$ -12 $":8624,"$ -4 $":8624,"$$ (t + 3)(t - 4) = 0 $$":8624,"$ t = -3 $":8624,"$ t = 4 $":8624,"$ 16 $":8624,"$ 1971 $":8624,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fhammer-feather.mp4":8654,"$ R_t $":8624,"$ R_1 $":8624,"$ R_2 $":8624,"$$ \\frac{1}{R_t} = \\frac{1}{R_1} + \\frac{1}{R_2} + \\dots $$":8624,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Fcircuit.svg":8656,"$$ R_2 = R_1 + 3 $$":8624,"$$ \\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_2} $$":8624,"$$ \\frac{1}{2} = \\frac{1}{R_1} + \\frac{1}{R_1 + 3} \\\\ R_1(R_1 + 3) = 2(R_1 + 3) + 2R_1 \\\\ R_1^2 + 3R_1 = 2R_1 + 6 + 2R_1 \\\\ R_1^2 + 3R_1 = 4R_1 + 6 \\\\ R_1^2 - R_1 - 6 = 0 $$":8624,"$ -6 $":8624,"$$ (R_1 - 3)(R_1 + 2) = 0 $$":8624,"$ R_1 = 3 $":8624,"$ R_1 = -2 $":8624,"$$ R_2 = R_1 + 3 = 3 + 3 = 6 $$":8624,"$ S = V \\cdot t $":8624,"$ V = 2 $":8624,"$ t = 10 $":8624,"$ S = V \\cdot t = 2 \\cdot 10 = 20 $":8624,"$$ S = V \\cdot t >>{big} V = \\frac{S}{t} >>{big} t = \\frac{S}{V} $$":8624,"content\u002F01-foundations\u002F02-equations\u002F03-quadratic\u002F6-real-world\u002Fassets\u002Friver-ride.svg":8660,"$ V $":8624,"$ V - 2 $":8624,"$$ t_{\\rarr} = \\frac{8}{V - 2} $$":8624,"$ V + 2 $":8624,"$$ t_{\\larr} = \\frac{8}{V + 2} $$":8624,"$$ t_{\\rarr} + t_{\\larr} = 3 $$":8624,"$$ \\frac{8}{V - 2} + \\frac{8}{V + 2} = 3 \\\\ 8(V + 2) + 8(V - 2) = 3(V - 2)(V + 2) \\\\ 8V + 16 + 8V - 16 = 3(V^2 - 4) \\\\ 16V = 3V^2 - 12 \\\\ 0 = 3V^2 - 16V - 12 $$":8624,"$$ D = (-16)^2 - 4 \\cdot 3 \\cdot (-12) = 256 + 144 = 400 $$":8624,"$$ V = \\frac{16 \\pm \\sqrt{400}}{2 \\cdot 3} = \\frac{16 \\pm 20}{6} $$":8624,"$ V = 6 $":8624,"$ V = -\\frac{2}{3} $":8624,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Freal-world\u002F$_whyNotReal":8664,"$$ t_{\\rarr} = \\frac{8}{6 - 2} = \\frac{8}{4} = 2 $$":8624,"$ 14 $":8624,"$ S = Vt $":8624,"$$ \\boxed{t_{\\text{upstream}} + t_{\\text{downstream}} = t_{\\text{still}}} $$":8624,"$$ S = Vt $$":8624,"$$ \\frac{S}{V} = \\frac{\\cancel{V}t}{\\cancel{V}} >>{big} \\boxed{t = \\frac{S}{V}} $$":8624,"$$ \\frac{S_{\\text{upstream}}}{V_{\\text{upstream}}} + \\frac{S_{\\text{downstream}}}{V_{\\text{downstream}}} = \\frac{S_{\\text{still}}}{V_{\\text{still}}} $$":8624,"$ V_б $":8624,"$ V_б + 1 $":8624,"$$ \\frac{15}{V_б - 1} + \\frac{14}{V_б + 1} = \\frac{30}{V_б} $$":8624,"$$ 15(V_б + 1)V_б + 14(V_б - 1)V_б = 30(V_б + 1)(V_б - 1) \\\\ 15(V_б^2 + V_б) + 14(V_б^2 - V_б) = 30(V_б^2 - 1) \\\\ 15V_б^2 + 15V_б + 14V_б^2 - 14V_б = 30V_б^2 - 30 \\\\ 29V_б^2 + V_б = 30V_б^2 - 30 \\\\ 0 = 30V_б^2 - 29V_б^2 - V_б - 30 \\\\ 0 = V_б^2 - V_б - 30 $$":8624,"$$ D = (-1)^2 - 4 \\cdot 1 \\cdot (-30) = 1 + 120 = 121 $$":8624,"$$ V_б = \\frac{-(-1) \\pm \\sqrt{121}}{2 \\cdot 1} = \\frac{1 \\pm 11}{2} $$":8624,"$ V_б = 6 $":8624,"$ V_б = -5 $":8624,"$ 200 $":8624,"$ V_1 $":8624,"$ t_1 $":8624,"$ V_2 = V_1 + 10 $":8624,"$ t_2 = t_1 - 1 $":8624,"$$ V_2 = V_1 + 10 $$":8624,"$ V = \\frac{S}{t} $":8624,"$$ \\frac{S}{t_2} = \\frac{S}{t_1} + 10 \\\\ \\frac{200}{t_1 - 1} = \\frac{200}{t_1} + 10 $$":8624,"$ t_1(t_1 - 1) $":8624,"$$ 200t_1 = 200(t_1 - 1) + 10t_1(t_1 - 1) \\\\ 200t_1 = 200t_1 - 200 + 10t_1^2 - 10t_1 \\\\ 0 = 10t_1^2 - 10t_1 - 200 \\\\ 0 = t_1^2 - t_1 - 20 $$":8624,"$ -20 $":8624,"$$ (t_1 + 4)(t_1 - 5) = 0 $$":8624,"$$ S = Vt >>{big} \\frac{S}{t} = V >>{big} V = \\frac{200}{5} = 40 $$":8624,"$ \\frac{S}{V} $":8624,"$ 1.5 $":8624,"$ t_{\\uarr} $":8624,"$ t_{\\darr} $":8624,"$ V_\\approx $":8624,"$$ t_{\\darr} = t_{\\uarr} - \\underbrace{1.5}_{\\small\\frac{3}{2}} $$":8624,"$$ \\frac{36}{10 + V_\\approx} = \\frac{36}{10 - V_\\approx} - \\frac{3}{2} \\\\ 36(10 - V_\\approx) = 36(10 + V_\\approx) - \\frac{3}{2}(10 + V_\\approx)(10 - V_\\approx) \\\\ \\cancel{360} - 36V_\\approx = \\cancel{360} + 36V_\\approx - \\frac{3}{2}(100 - V_\\approx^2) \\\\ -72V_\\approx = - \\frac{3}{2}(100 - V_\\approx^2) \\\\ -24V_\\approx = -\\frac{1}{2}(100 - V_\\approx^2) \\\\ 48V_\\approx = 100 - V_\\approx^2 \\\\ V_\\approx^2 + 48V_\\approx - 100 = 0 $$":8624,"$$ (V_\\approx + 50)(V_\\approx - 2) = 0 $$":8624,"$ -50 $":8624,"$ t_2 = t_1 - 2 $":8624,"$$ V_2 = V_1 + 1 $$":8624,"$$ \\frac{40}{t_2} = \\frac{40}{t_1} + 1 $$":8624,"$ t_2 $":8624,"$$ \\frac{40}{t_1 - 2} = \\frac{40}{t_1} + 1 $$":8624,"$ t_1(t_1 - 2) $":8624,"$$ 40t_1 = 40(t_1 - 2) + t_1(t_1 - 2) \\\\ 40t_1 = 40t_1 - 80 + t_1^2 - 2t_1 \\\\ 0 = t_1^2 - 2t_1 - 80 $$":8624,"$ -80 $":8624,"$ -10 $":8624,"$$ (t_1 + 8)(t_1 - 10) = 0 $$":8624,"$ t_1 = -8 $":8624,"$ t_1 = 10 $":8624,"$ 18 $":8624,"$ 24 $":8624,"$ V + 3 $":8624,"$ 2V $":8624,"$ 2(V + 3) = 2V + 6 $":8624,"$ 2V + 6 $":8624,"$$ (2V)^2 + (2V + 6)^2 = 30^2 $$":8624,"$$ 4V^2 + 4V^2 + 24V + 36 = 900 \\\\ 8V^2 + 24V + 36 = 900 \\\\ 8V^2 + 24V - 864 = 0 $$":8624,"$$ V^2 + 3V - 108 = 0 $$":8624,"$ -108 $":8624,"$ -9 $":8624,"$$ (V + 12)(V - 9) = 0 $$":8624,"$ V = -12 $":8624,"$ V = 9 $":8624,"$ 9 $":8624,"$$ d_{\\text{east}} = 2 \\cdot 9 = 18 >>{big} d_{\\text{north}} = 2 \\cdot 12 = 24 $$":8624,"$ S $":8624,"$ W $":8624,"$ P = 8 $":8624,"$ 8 \\cdot 10 = 80 = W $":8624,"$ W = Pt $":8624,"$$ W = Pt >>{big} P = \\frac{W}{t} >>{big} t = \\frac{W}{P} $$":8624,"$$ W = (P_1 + P_2) \\cdot 6 $$":8624,"$ P = \\frac{W}{t} $":8624,"$$ W = \\left(\\frac{W}{t_1} + \\frac{W}{t_2}\\right) \\cdot 6 $$":8624,"$$ \\cancel{W} = \\cancel{W} \\cdot \\left(\\frac{1}{t_1} + \\frac{1}{t_2}\\right) \\cdot 6 \\\\ 1 = \\left(\\frac{1}{t_1} + \\frac{1}{t_2}\\right) \\cdot 6 \\\\ \\frac{1}{6} = \\frac{1}{t_1} + \\frac{1}{t_2} $$":8624,"$ t_1 = t_2 - 5 $":8624,"$$ \\frac{1}{6} = \\frac{1}{t_2 - 5} + \\frac{1}{t_2} $$":8624,"$$ \\frac{(t_2 - 5)t_2}{6} = t_2 + t_2 - 5 \\\\ \\frac{t_2^2 - 5t_2}{6} = 2t_2 - 5 \\\\ t_2^2 - 5t_2 = 12t_2 - 30 \\\\ t_2^2 - 17t_2 + 30 = 0 $$":8624,"$ -15 $":8624,"$ -17 $":8624,"$$ (t_2 - 15)(t_2 - 2) = 0 $$":8624,"$ t_1 = t_2 - 5 = -3 $":8624,"$ 240 $":8624,"$ 35% $":8624,"$ 27 $":8624,"$ W_1 = 240 $":8624,"$$ W_2 = W_1 + \\frac{35}{100} \\cdot W_1 = 240 + \\frac{35}{100} \\cdot 240 = 324 $$":8624,"$$ P_1 = P_2 - 3 >>{big} t_1 = t_2 - 2 $$":8624,"$$ \\frac{W_1}{P_1} = \\frac{W_2}{P_2} - 2 $$":8624,"$ P_1 $":8624,"$ P_2 $":8624,"$$ \\frac{240}{P_2 - 3} = \\frac{324}{P_2} - 2 \\\\ 240P_2 = 324(P_2 - 3) - 2P_2(P_2 - 3) \\\\ 120P_2 = 162(P_2 - 3) - P_2(P_2 - 3) \\\\ 120P_2 = 162P_2 - 486 - P_2^2 + 3P_2 \\\\ P_2^2 - 45P_2 + 486 = 0 $$":8624,"$$ D = (-45)^2 - 4 \\cdot 1 \\cdot 486 = 2025 - 1944 = 81 $$":8624,"$$ P_2 = \\frac{-(-45) \\pm \\sqrt{81}}{2 \\cdot 1} = \\frac{45 \\pm 9}{2} $$":8624,"$ P_2 = 27 $":8624,"$ P_2 = 18 $":8624,"$ P_1 = 27 - 3 = 24 $":8624,"$ 3\u002F4 $":8624,"$ \\frac{56}{3} $":8624,"$$ W = (P_1 + P_2 + P_3) \\cdot 6 $$":8624,"$$ t_2 = \\frac{3}{4}t_1 >>{big} t_3 = t_2 + 10 $$":8624,"$$ W = \\left(\\frac{W}{t_1} + \\frac{W}{t_2} + \\frac{W}{t_3}\\right) \\cdot 6 $$":8624,"$$ 1 = \\left(\\frac{1}{t_1} + \\frac{1}{t_2} + \\frac{1}{t_3}\\right) \\cdot 6 $$":8624,"$$ t_2 = \\frac{3}{4}t_1 \\implies t_1 = \\frac{4}{3}t_2 $$":8624,"$ t_3 = t_2 + 10 $":8624,"$$ 1 = \\left(\\frac{1}{\\frac{4}{3}t_2} + \\frac{1}{t_2} + \\frac{1}{t_2 + 10}\\right) \\cdot 6 \\\\ \\frac{1}{6} = \\frac{3}{4t_2} + \\frac{1}{t_2} + \\frac{1}{t_2 + 10} \\\\ \\frac{1}{6} = \\frac{7}{4t_2} + \\frac{1}{t_2 + 10} $$":8624,"$ 12t_2(t_2 + 10) $":8624,"$$ 2t_2(t_2 + 10) = 21(t_2 + 10) + 12t_2 \\\\ 2t_2^2 + 20t_2 = 21t_2 + 210 + 12t_2 \\\\ 2t_2^2 - 13t_2 - 210 = 0 $$":8624,"$$ D = (-13)^2 - 4 \\cdot 2 \\cdot (-210) = 169 + 1680 = 1849 $$":8624,"$$ t_2 = \\frac{-(-13) \\pm \\sqrt{1849}}{2 \\cdot 2} = \\frac{13 \\pm 43}{4} $$":8624,"$ t_2 = 14 $":8624,"$ t_2 = -7.5 $":8624,"$$ t_1 = \\frac{4}{3} \\cdot 14 = \\frac{56}{3} \\text{ min} >>{big} t_3 = 14 + 10 = 24 \\text{ min} $$":8624,"$ c $":8624,"$$ c_A = \\frac{m_A}{M} >>{big} c_A = \\frac{V_A}{V} $$":8624,"$ 3% $":8624,"$ c = 0.03 $":8624,"$ 600 $":8624,"$ 4% $":8624,"$ 0.06 $":8624,"$ 6% $":8624,"$ M $":8624,"$$ c = \\frac{18}{M} $$":8624,"$ 0.04 $":8624,"$$ c - 0.04 = \\frac{18}{M + 600} $$":8624,"$$ \\frac{18}{M} - 0.04 = \\frac{18}{M + 600} \\\\ \\frac{18}{M} - \\frac{4}{100} = \\frac{18}{M + 600} \\\\ \\frac{9}{M} - \\frac{2}{100} = \\frac{9}{M + 600} \\\\ 900(M+600) - 2M(M+600) = 900M \\\\ 450(M+600) - M(M+600) = 450M \\\\ \\cancel{450M} + 450 \\cdot 600 - M^2 - 600M = \\cancel{450M} \\\\ 450 \\cdot 600 - M^2 - 600M = 0 \\\\ M^2 + 600M - 450 \\cdot 600 = 0 $$":8624,"$$ M = -300 \\pm \\sqrt{300^2 + 450 \\cdot 600} = -300 \\pm 600 $$":8624,"$ M = 300 $":8624,"$ M = -900 $":8624,"$$ c = \\frac{18}{300} = \\frac{6}{100} = 0.06 $$":8624,"$ 40% $":8624,"$ 36% $":8624,"$ m_1 + m_2 = M $":8624,"$ m_1 $":8624,"$ m_2 $":8624,"$ 20% $":8624,"$$ 0.36 = \\frac{6 + 12}{M} \\implies M = \\frac{18}{0.36} = \\frac{18 \\cdot 100}{36} = 50 $$":8624,"$$ c_1 = c_2 - 0.4 $$":8624,"$$ m_1 + m_2 = 50 $$":8624,"$$ c_1 = \\frac{6}{m_1} \\implies m_1 = \\frac{6}{c_1} >>{big} c_2 = \\frac{12}{m_2} \\implies m_2 = \\frac{12}{c_2} $$":8624,"$$ \\frac{6}{c_1} + \\frac{12}{c_2} = 50 $$":8624,"$$ \\frac{6}{c_2 - 0.4} + \\frac{12}{c_2} = 50 $$":8624,"$$ \\frac{6}{c_2 - \\frac{4}{10}} + \\frac{12}{c_2} = 50 \\\\ \\frac{3}{c_2 - \\frac{4}{10}} + \\frac{6}{c_2} = 25 \\\\ 3c_2 + 6\\left(c_2 - \\frac{4}{10}\\right) = 25c_2\\left(c_2 - \\frac{4}{10}\\right) \\\\ 3c_2 + 6c_2 - \\frac{24}{10} = 25c_2^2 - 10c_2 \\\\ 0 = 25c_2^2 - 19c_2 + \\frac{12}{5} $$":8624,"$$ D = (-19)^2 - 4 \\cdot 25 \\cdot \\frac{12}{5} = 361 - 240 = 121 $$":8624,"$$ c_2 = \\frac{-(-19) \\pm \\sqrt{121}}{2 \\cdot 25} = \\frac{19 \\pm 11}{50} $$":8624,"$ c_2 = 0.6 $":8624,"$ c_2 = 0.16 $":8624,"$ c_1 = -0.24 $":8624,"$ c_1 = 0.2 $":8624,"$ 16% $":8624,"$ 9% $":8624,"$ 8-x $":8624,"$ (8-x) \\cdot 0.16 $":8624,"$$ \\frac{(8-x) \\cdot 0.16}{8} = (8-x) \\cdot 0.02 $$":8624,"$ 0.16 - 0.02x $":8624,"$ (8-x) \\cdot (8-x) \\cdot 0.02 $":8624,"$ 0.09 $":8624,"$$ \\frac{(8-x) \\cdot (8-x) \\cdot 0.02}{8} = 0.09 \\\\ \\frac{(8-x)^2 \\cdot 0.02}{8} = 0.09 \\\\ $$":8624,"$ 100 $":8624,"$$ \\frac{(8-x)^2 \\cdot 2}{8} = 9 \\\\ (8-x)^2 = 36 \\\\ 8-x = \\pm 6 \\\\ 8\\mp 6 = x $$":8624,"$ x = 14 $":8624,"$ n $":8624,"$ 1, 2, 3\\ldots $":8624,"$$ S = \\frac{1}{2}n(n+1) $$":8624,"$ 465 $":8624,"$$ 465 = \\frac{1}{2}n(n+1) \\\\ 930 = n(n+1) \\\\ 930 = n^2 + n \\\\ 0 = n^2 + n - 930 $$":8624,"$$ D = 1^2 - 4 \\cdot 1 \\cdot (-930) = 1 + 3720 = 3721 $$":8624,"$$ n = \\frac{-1 \\pm \\sqrt{3721}}{2 \\cdot 1} = \\frac{-1 \\pm 61}{2} $$":8624,"$ n = 30 $":8624,"$ n = -31 $":8624,"$ -31 $":8624,"$ 22 $":8624,"$ m $":8624,"$$ n + m = 22 >>{big} n\\cdot m = 120 $$":8624,"$$ n = 22 - m $$":8624,"$$ (22 - m)m = 120 \\\\ 22m - m^2 = 120 \\\\ 0 = m^2 - 22m + 120 $$":8624,"$ A = 1 $":8624,"$ -22 $":8624,"$$ (m - 10)(m - 12) = 0 $$":8624,"$ m = 10 $":8624,"$ m = 12 $":8624,"$ n = 22 - m $":8624,"$$ 12 = 1\\cdot 10 + 2 >>{big} 84 = 8\\cdot 10 + 4 $$":8624,"$ 26 $":8624,"$ b $":8624,"$ 1\\cdot 10 + 2 $":8624,"$ 84 $":8624,"$ 8\\cdot 10 + 4 $":8624,"$$ ab = 12 $$":8624,"$$ 10a + b + 36 = 10b + a \\\\ 9a + 36 = 9b \\\\ a + 4 = b $$":8624,"$ ab = 12 $":8624,"$$ a(a + 4) = 12 \\\\ a^2 + 4a - 12 = 0 $$":8624,"$$ (a + 6)(a - 2) = 0 $$":8624,"$ a = -6 $":8624,"$ a = 2 $":8624,"$ b = a + 4 = 6 $":8624,"$ 46 $":8624,"$ 17 $":8624,"$$ a + b + a + b = 46 \\\\ 2a + 2b = 46 $$":8624,"$$ a + b = 23 $$":8624,"$$ a^2 + b^2 = 17^2 \\\\ a^2 + b^2 = 289 $$":8624,"$ a + b = 23 $":8624,"$ a = 23 - b $":8624,"$$ (23 - b)^2 + b^2 = 289 \\\\ 529 - 46b + b^2 + b^2 = 289 \\\\ 2b^2 - 46b + 240 = 0 $$":8624,"$$ b^2 - 23b + 120 = 0 $$":8624,"$$ D = (-23)^2 - 4 \\cdot 1 \\cdot 120 = 529 - 480 = 49 \\\\ b = \\frac{-(-23) \\pm \\sqrt{49}}{2 \\cdot 1} = \\frac{23 \\pm 7}{2} $$":8624,"$ b = 15 $":8624,"$ b = 8 $":8624,"$ a = 8 $":8624,"$ a = 15 $":8624,"$ 41 $":8624,"$$ a^2 + b^2 = c^2 $$":8624,"$$ a^2 + b^2 = 5^2 \\\\ a^2 + b^2 = 25 $$":8624,"$$ (3a)^2 + (10b)^2 = 41^2 \\\\ 9a^2 + 100b^2 = 1681 $$":8624,"$ a^2 $":8624,"$ b^2 $":8624,"$ a^2 = 25 - b^2 $":8624,"$$ 9(25 - b^2) + 100b^2 = 1681 \\\\ 225 - 9b^2 + 100b^2 = 1681 \\\\ 91b^2 = 1681 - 225 \\\\ 91b^2 = 1456 \\\\ b^2 = 16 \\\\ b = \\pm 4 $$":8624,"$$ a^2 + 4^2 = 25 \\\\ a^2 + 16 = 25 \\\\ a^2 = 9 \\\\ a = \\pm 3 $$":8624,"$ 50 $":8624,"$$ a \\cdot b = 50 >>{big}{bottom} a + b + a + b = 30 \\\\ 2(a+b) = 30 $$":8624,"$ a = 50\u002Fb $":8624,"$$ 2\\left(\\frac{50}{b} + b\\right) = 30 \\\\ \\frac{100}{b} + 2b = 30 \\\\ 100 + 2b^2 = 30b \\\\ 2b^2 - 30b + 100 = 0 \\\\ b^2 - 15b + 50 = 0 $$":8624,"$$ (b - 10)(b - 5) = 0 $$":8624,"$ b = 10 $":8624,"$ b = 5 $":8624,"$$ a = \\frac{50}{10} = 5 >>{big} a = \\frac{50}{5} = 10 $$":8624,"$ 28 $":8624,"$$ \\frac{a}{b} = \\frac{5}{4} $$":8624,"$ (a - 2)^2 $":8624,"$ (b - 2)^2 $":8624,"$$ a = \\frac{5}{4}b $$":8624,"$$ (a - 2)^2 - (b - 2)^2 = 28 $$":8624,"$$ (a - 2 - (b - 2))(a - 2 + (b - 2)) = 28 \\\\ (a - b)(a + b - 4) = 28 $$":8624,"$ \\frac{5}{4}b $":8624,"$$ \\left(\\frac{5}{4}b - b\\right)\\left(\\frac{5}{4}b + b - 4\\right) = 28 \\\\ \\left(\\frac{1}{4}b\\right)\\left(\\frac{9}{4}b - 4\\right) = 28 \\\\ \\frac{9}{16}b^2 - b = 28 \\\\ 9b^2 - 16b = 448 \\\\ 9b^2 - 16b - 448 = 0 $$":8624,"$$ b = \\frac{-(-8) \\pm \\sqrt{(-8)^2 - 9 \\cdot (-448)}}{9} = \\frac{8 \\pm \\sqrt{64 + 4032}}{9} = \\frac{8 \\pm \\sqrt{4096}}{9} = \\frac{8 \\pm 64}{9} $$":8624,"$ b = -\\frac{56}{9} $":8624,"$$ a = \\frac{5}{4}b = \\frac{5}{4} \\cdot 8 = 10 $$":8624,"$$ a = 5x >>{big} b = 4y >>{big} x \\neq y $$":8624,"$ a : b = 5 : 4 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