{"id":3149,"date":"2025-08-15T22:18:31","date_gmt":"2025-08-15T22:18:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3149"},"modified":"2026-01-14T19:51:54","modified_gmt":"2026-01-14T19:51:54","slug":"variation-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/variation-apply-it\/","title":{"raw":"Variation: Apply It","rendered":"Variation: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve direct variation problems.<\/li>\r\n \t<li>Solve inverse variation problems.<\/li>\r\n \t<li>Solve problems involving joint variation.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Animal Kingdom Adventures<\/h2>\r\n<p class=\"whitespace-normal break-words\">Welcome to the world of animal biology! From the fastest cheetahs to the largest elephants, the animal kingdom is full of fascinating mathematical relationships. Zoologists, wildlife photographers, and conservationists use variation models to understand animal behavior, predict population changes, and design better wildlife habitats.<\/p>\r\n\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">The Great Migration - Cheetah Speed Analysis<\/h3>\r\n<p class=\"whitespace-normal break-words\">Suppose that wildlife researchers have discovered that a cheetah's hunting success rate varies directly with the square of its sprint speed. When a cheetah runs at 60 mph, it has an 85% success rate catching gazelles.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Find the speed-success relationship and predict hunting outcomes at different speeds:<\/p>\r\n\r\n<ul>\r\n \t<li>45mph<\/li>\r\n \t<li>70mph<\/li>\r\n \t<li>30mph<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"114753\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"114753\"]\r\n\r\nSince success rate varies directly with the square of speed, we use [latex]S = kv^2[\/latex], where [latex]S[\/latex] is success rate (as a percentage) and [latex]v[\/latex] is speed in mph.\r\n\r\n[latex]\\begin{aligned} 85 &amp;= k(60)^2 \\quad \\text{substitute known values} \\\\ 85 &amp;= 3600k \\\\ k &amp;= \\frac{85}{3600} \\approx 0.0236 \\end{aligned}[\/latex]\r\n\r\nThe hunting success formula is [latex]S = 0.0236v^2[\/latex].\r\n\r\nLet's predict success rates at different speeds:\r\n<ul>\r\n \t<li>At 45 mph:\r\n[latex]S = 0.0236(45)^2 \\approx 47.8%[\/latex]<\/li>\r\n \t<li>At 70 mph:\r\n[latex]S = 0.0236(70)^2 \\approx 115.6%[\/latex] (theoretical maximum is 100%)<\/li>\r\n \t<li>At 30 mph:\r\n[latex]S = 0.0236(30)^2 \\approx 21.2%[\/latex][\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Elephant Water Needs<\/h3>\r\n<p class=\"whitespace-normal break-words\">Suppose that a different set of researchers found that the daily water consumption of elephants varies inversely with the square of the distance they must travel to reach water sources. When the watering hole is 4 kilometers away, elephants drink 150 gallons per day.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Create a water consumption model for different distances to water sources.<\/p>\r\n\r\n<ul>\r\n \t<li>At 2km<\/li>\r\n \t<li>At 6km<\/li>\r\n \t<li>At 8km<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"17952\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"17952\"]\r\n\r\nSince water consumption varies inversely with the square of distance, we use [latex]W = \\frac{k}{d^2}[\/latex].\r\n\r\n[latex]\\begin{aligned} 150 &amp;= \\frac{k}{4^2} \\\\ 150 &amp;= \\frac{k}{16} \\\\ k &amp;= 2,400 \\end{aligned}[\/latex]\r\n\r\nThe water consumption formula is [latex]W = \\frac{2,400}{d^2}[\/latex].\r\n\r\nWater consumption at different distances:\r\n<ul>\r\n \t<li>At 2 km:\r\n[latex]W = \\frac{2,400}{2^2} = 600[\/latex] gallons\/day<\/li>\r\n \t<li>At 6 km:\r\n[latex]W = \\frac{2,400}{6^2} \\approx 66.7[\/latex] gallons\/day<\/li>\r\n \t<li>At 8 km:\r\n[latex]W = \\frac{2,400}{8^2} = 37.5[\/latex] gallons\/day[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Penguin Colony Dynamics<\/h3>\r\n<p class=\"whitespace-normal break-words\">Now, suppose that marine biologists studying emperor penguins in Antarctica discovered that fish catch rates vary in a complex way. The number of fish caught per hour [latex]F[\/latex] varies directly with the square of the number of penguins in a hunting group [latex]P[\/latex] and inversely with the depth [latex]D[\/latex] they must dive (in meters).<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">When 6 penguins hunt together at a depth of 18 meters, they catch 24 fish per hour. Find the model and predict different scenarios.<\/p>\r\n\r\n<ul>\r\n \t<li>4 penguins at 12m depth<\/li>\r\n \t<li>8 penguins at 24m depth<\/li>\r\n \t<li>3 penguins at 9m depth<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">[reveal-answer q=\"513579\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"513579\"]<\/p>\r\n<p class=\"whitespace-normal break-words\">The joint variation formula is [latex]F = \\frac{kP^2}{D}[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]\\begin{aligned} 24 &amp;= \\frac{k(6)^2}{18} \\\\ 24 &amp;= \\frac{36k}{18} \\\\ 24 &amp;= 2k \\\\ k &amp;= 12 \\end{aligned}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The fishing formula is [latex]F = \\frac{12P^2}{D}[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">Testing different hunting scenarios:<\/p>\r\n\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">4 penguins at 12m depth:\r\n[latex]F = \\frac{12(4)^2}{12} = \\frac{192}{12} = 16[\/latex] fish\/hour<\/li>\r\n \t<li class=\"whitespace-normal break-words\">8 penguins at 24m depth:\r\n[latex]F = \\frac{12(8)^2}{24} = \\frac{768}{24} = 32[\/latex] fish\/hour<\/li>\r\n \t<li class=\"whitespace-normal break-words\">3 penguins at 9m depth:\r\n[latex]F = \\frac{12(3)^2}{9} = \\frac{108}{9} = 12[\/latex] fish\/hour[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox youChoose\" aria-label=\"You Choose\">[choosedataset divId=\"tnh-choose-dataset\" title=\"Choose Your Own Animal\" label=\"\" default=\"Choose an Animal\"]\r\n[datasetoption]\r\n[displayname]Dolphins[\/displayname]\r\n[ohmid]42381[\/ohmid]\r\n[\/datasetoption][datasetoption]\r\n[displayname]Turtles[\/displayname]\r\n[ohmid]42382[\/ohmid]\r\n[\/datasetoption][datasetoption]\r\n[displayname]Bats[\/displayname]\r\n[ohmid]42383[\/ohmid]\r\n[\/datasetoption]\r\n[\/choosedataset]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve direct variation problems.<\/li>\n<li>Solve inverse variation problems.<\/li>\n<li>Solve problems involving joint variation.<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Animal Kingdom Adventures<\/h2>\n<p class=\"whitespace-normal break-words\">Welcome to the world of animal biology! From the fastest cheetahs to the largest elephants, the animal kingdom is full of fascinating mathematical relationships. Zoologists, wildlife photographers, and conservationists use variation models to understand animal behavior, predict population changes, and design better wildlife habitats.<\/p>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">The Great Migration &#8211; Cheetah Speed Analysis<\/h3>\n<p class=\"whitespace-normal break-words\">Suppose that wildlife researchers have discovered that a cheetah&#8217;s hunting success rate varies directly with the square of its sprint speed. When a cheetah runs at 60 mph, it has an 85% success rate catching gazelles.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Find the speed-success relationship and predict hunting outcomes at different speeds:<\/p>\n<ul>\n<li>45mph<\/li>\n<li>70mph<\/li>\n<li>30mph<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q114753\">Show Solution<\/button><\/p>\n<div id=\"q114753\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since success rate varies directly with the square of speed, we use [latex]S = kv^2[\/latex], where [latex]S[\/latex] is success rate (as a percentage) and [latex]v[\/latex] is speed in mph.<\/p>\n<p>[latex]\\begin{aligned} 85 &= k(60)^2 \\quad \\text{substitute known values} \\\\ 85 &= 3600k \\\\ k &= \\frac{85}{3600} \\approx 0.0236 \\end{aligned}[\/latex]<\/p>\n<p>The hunting success formula is [latex]S = 0.0236v^2[\/latex].<\/p>\n<p>Let&#8217;s predict success rates at different speeds:<\/p>\n<ul>\n<li>At 45 mph:<br \/>\n[latex]S = 0.0236(45)^2 \\approx 47.8%[\/latex]<\/li>\n<li>At 70 mph:<br \/>\n[latex]S = 0.0236(70)^2 \\approx 115.6%[\/latex] (theoretical maximum is 100%)<\/li>\n<li>At 30 mph:<br \/>\n[latex]S = 0.0236(30)^2 \\approx 21.2%[\/latex]<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Elephant Water Needs<\/h3>\n<p class=\"whitespace-normal break-words\">Suppose that a different set of researchers found that the daily water consumption of elephants varies inversely with the square of the distance they must travel to reach water sources. When the watering hole is 4 kilometers away, elephants drink 150 gallons per day.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Create a water consumption model for different distances to water sources.<\/p>\n<ul>\n<li>At 2km<\/li>\n<li>At 6km<\/li>\n<li>At 8km<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q17952\">Show Solution<\/button><\/p>\n<div id=\"q17952\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since water consumption varies inversely with the square of distance, we use [latex]W = \\frac{k}{d^2}[\/latex].<\/p>\n<p>[latex]\\begin{aligned} 150 &= \\frac{k}{4^2} \\\\ 150 &= \\frac{k}{16} \\\\ k &= 2,400 \\end{aligned}[\/latex]<\/p>\n<p>The water consumption formula is [latex]W = \\frac{2,400}{d^2}[\/latex].<\/p>\n<p>Water consumption at different distances:<\/p>\n<ul>\n<li>At 2 km:<br \/>\n[latex]W = \\frac{2,400}{2^2} = 600[\/latex] gallons\/day<\/li>\n<li>At 6 km:<br \/>\n[latex]W = \\frac{2,400}{6^2} \\approx 66.7[\/latex] gallons\/day<\/li>\n<li>At 8 km:<br \/>\n[latex]W = \\frac{2,400}{8^2} = 37.5[\/latex] gallons\/day<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Penguin Colony Dynamics<\/h3>\n<p class=\"whitespace-normal break-words\">Now, suppose that marine biologists studying emperor penguins in Antarctica discovered that fish catch rates vary in a complex way. The number of fish caught per hour [latex]F[\/latex] varies directly with the square of the number of penguins in a hunting group [latex]P[\/latex] and inversely with the depth [latex]D[\/latex] they must dive (in meters).<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">When 6 penguins hunt together at a depth of 18 meters, they catch 24 fish per hour. Find the model and predict different scenarios.<\/p>\n<ul>\n<li>4 penguins at 12m depth<\/li>\n<li>8 penguins at 24m depth<\/li>\n<li>3 penguins at 9m depth<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q513579\">Show Solution<\/button><\/p>\n<div id=\"q513579\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-normal break-words\">The joint variation formula is [latex]F = \\frac{kP^2}{D}[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">[latex]\\begin{aligned} 24 &= \\frac{k(6)^2}{18} \\\\ 24 &= \\frac{36k}{18} \\\\ 24 &= 2k \\\\ k &= 12 \\end{aligned}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The fishing formula is [latex]F = \\frac{12P^2}{D}[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">Testing different hunting scenarios:<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">4 penguins at 12m depth:<br \/>\n[latex]F = \\frac{12(4)^2}{12} = \\frac{192}{12} = 16[\/latex] fish\/hour<\/li>\n<li class=\"whitespace-normal break-words\">8 penguins at 24m depth:<br \/>\n[latex]F = \\frac{12(8)^2}{24} = \\frac{768}{24} = 32[\/latex] fish\/hour<\/li>\n<li class=\"whitespace-normal break-words\">3 penguins at 9m depth:<br \/>\n[latex]F = \\frac{12(3)^2}{9} = \\frac{108}{9} = 12[\/latex] fish\/hour<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox youChoose\" aria-label=\"You Choose\">\n<div id=\"tnh-choose-dataset\" class=\"chooseDataset\">\n<h3>Choose Your Own Animal<\/h3>\n<form><select name=\"dataset\"><option value=\"\">Choose an Animal<\/option><option value=\"42381\">Dolphins<\/option><option value=\"42382\">Turtles<\/option><option value=\"42383\">Bats<\/option><\/select><\/form>\n<div class=\"ohmContainer\"><\/div>\n<\/p><\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":508,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":[{"divId":"tnh-choose-dataset","title":"Choose Your Own Animal","label":"","default":"Choose an Animal","try_it_collection":[{"displayName":"Dolphins","value":"42381"},{"displayName":"Turtles","value":"42382"},{"displayName":"Bats","value":"42383"}]}],"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3149"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3149\/revisions"}],"predecessor-version":[{"id":5366,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3149\/revisions\/5366"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/508"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3149\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3149"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3149"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3149"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}