{"id":3121,"date":"2024-06-12T19:22:34","date_gmt":"2024-06-12T19:22:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3121"},"modified":"2024-08-05T02:28:01","modified_gmt":"2024-08-05T02:28:01","slug":"applied-optimization-problems-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/applied-optimization-problems-fresh-take\/","title":{"raw":"Applied Optimization Problems: Fresh Take","rendered":"Applied Optimization Problems: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Tackle business problems to find the best ways to increase profits, minimize costs, or maximize revenue<\/li>\r\n\t<li>Use optimization methods to solve problems involving geometry<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Optimization Problems<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Optimization Goal:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Find maximum or minimum values of functions under certain constraints<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Types of Intervals:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Closed and bounded: Guaranteed to have absolute extrema<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Open or unbounded: May or may not have absolute extrema<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Critical Points:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Points where [latex]f'(x) = 0[\/latex] or [latex]f'(x)[\/latex] is undefined<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key to finding potential extrema<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Constraint Equations:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Used to express the problem in terms of a single variable<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Often in the form [latex]g(x,y) = k[\/latex], where [latex]k[\/latex] is a constant<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Domain Analysis:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Crucial for determining where to search for extrema<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Consider endpoints for closed intervals<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Analyze behavior as [latex]x \\to \\infty[\/latex] or [latex]x \\to -\\infty[\/latex] for unbounded intervals<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Applications:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Geometry: Maximizing areas or volumes, minimizing surface areas<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Business: Maximizing revenue or profit, minimizing costs<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Physics: Minimizing energy, maximizing distances<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>In the Learn It pages, we gave the example: \"An open-top box is to be made from a [latex]24[\/latex] in. by [latex]36[\/latex] in. piece of cardboard by removing a square from each corner of the box and folding up the flaps on each side. What size square should be cut out of each corner to get a box with the maximum volume?\"<\/p>\r\n<p id=\"fs-id1165043108738\">Suppose the dimensions of the cardboard are now 20 in. by 30 in. Let [latex]x[\/latex] be the side length of each square and write the volume of the open-top box as a function of [latex]x[\/latex]. Determine the domain of consideration for [latex]x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"8044661\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"8044661\"]<\/p>\r\n<p id=\"fs-id1165043080382\">The volume of the box is [latex]L \\cdot W \\cdot H[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043177823\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043177823\"]<\/p>\r\n<p id=\"fs-id1165043177823\">[latex]V(x)=x(20-2x)(30-2x)[\/latex]. The domain is [latex][0,10][\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165043348922\">Suppose the island is [latex]1[\/latex] mi from shore, and the distance from the cabin to the point on the shore closest to the island is [latex]15[\/latex] mi. Suppose a visitor swims at the rate of [latex]2.5[\/latex] mph and runs at a rate of [latex]6[\/latex] mph. Let [latex]x[\/latex] denote the distance the visitor will run before swimming, and find a function for the time it takes the visitor to get from the cabin to the island.<\/p>\r\n<p>[reveal-answer q=\"212787\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"212787\"]<\/p>\r\n<p>The time [latex]T=T_{\\text{running}}+T_{\\text{swimming}}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042369705\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042369705\"]<\/p>\r\n<p id=\"fs-id1165042369705\">[latex]T(x)=\\frac{x}{6}+\\frac{\\sqrt{(15-x)^2+1}}{2.5}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165042376530\">A car rental company charges its customers [latex]p[\/latex] dollars per day, where [latex]60\\le p\\le 150[\/latex]. It has found that the number of cars rented per day can be modeled by the linear function [latex]n(p)=750-5p[\/latex]. How much should the company charge each customer to maximize revenue?<\/p>\r\n<p>[reveal-answer q=\"13435467\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"13435467\"]<\/p>\r\n<p id=\"fs-id1165042321515\">[latex]R(p)=n \\times p[\/latex], where [latex]n[\/latex] is the number of cars rented and [latex]p[\/latex] is the price charged per car.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042960136\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042960136\"]<\/p>\r\n<p id=\"fs-id1165042960136\">The company should charge [latex]$75[\/latex] per car per day.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165043199268\">Modify the area function [latex]A[\/latex] if the rectangle is to be inscribed in the unit circle [latex]x^2+y^2=1[\/latex]. What is the domain of consideration?<\/p>\r\n<p>[reveal-answer q=\"70845661\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"70845661\"]<\/p>\r\n<p id=\"fs-id1165042318676\">If [latex](x,y)[\/latex] is the vertex of the square that lies in the first quadrant, then the area of the square is [latex]A=(2x)(2y)=4xy[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043276112\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043276112\"]<\/p>\r\n<p id=\"fs-id1165043276112\">[latex]A(x)=4x\\sqrt{1-x^2}[\/latex]. The domain of consideration is [latex][0,1][\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Tackle business problems to find the best ways to increase profits, minimize costs, or maximize revenue<\/li>\n<li>Use optimization methods to solve problems involving geometry<\/li>\n<\/ul>\n<\/section>\n<h2>Solving Optimization Problems<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Optimization Goal:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find maximum or minimum values of functions under certain constraints<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Intervals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Closed and bounded: Guaranteed to have absolute extrema<\/li>\n<li class=\"whitespace-normal break-words\">Open or unbounded: May or may not have absolute extrema<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Critical Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Points where [latex]f'(x) = 0[\/latex] or [latex]f'(x)[\/latex] is undefined<\/li>\n<li class=\"whitespace-normal break-words\">Key to finding potential extrema<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Constraint Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used to express the problem in terms of a single variable<\/li>\n<li class=\"whitespace-normal break-words\">Often in the form [latex]g(x,y) = k[\/latex], where [latex]k[\/latex] is a constant<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain Analysis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Crucial for determining where to search for extrema<\/li>\n<li class=\"whitespace-normal break-words\">Consider endpoints for closed intervals<\/li>\n<li class=\"whitespace-normal break-words\">Analyze behavior as [latex]x \\to \\infty[\/latex] or [latex]x \\to -\\infty[\/latex] for unbounded intervals<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Geometry: Maximizing areas or volumes, minimizing surface areas<\/li>\n<li class=\"whitespace-normal break-words\">Business: Maximizing revenue or profit, minimizing costs<\/li>\n<li class=\"whitespace-normal break-words\">Physics: Minimizing energy, maximizing distances<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>In the Learn It pages, we gave the example: &#8220;An open-top box is to be made from a [latex]24[\/latex] in. by [latex]36[\/latex] in. piece of cardboard by removing a square from each corner of the box and folding up the flaps on each side. What size square should be cut out of each corner to get a box with the maximum volume?&#8221;<\/p>\n<p id=\"fs-id1165043108738\">Suppose the dimensions of the cardboard are now 20 in. by 30 in. Let [latex]x[\/latex] be the side length of each square and write the volume of the open-top box as a function of [latex]x[\/latex]. Determine the domain of consideration for [latex]x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8044661\">Hint<\/button><\/p>\n<div id=\"q8044661\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043080382\">The volume of the box is [latex]L \\cdot W \\cdot H[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043177823\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043177823\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043177823\">[latex]V(x)=x(20-2x)(30-2x)[\/latex]. The domain is [latex][0,10][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165043348922\">Suppose the island is [latex]1[\/latex] mi from shore, and the distance from the cabin to the point on the shore closest to the island is [latex]15[\/latex] mi. Suppose a visitor swims at the rate of [latex]2.5[\/latex] mph and runs at a rate of [latex]6[\/latex] mph. Let [latex]x[\/latex] denote the distance the visitor will run before swimming, and find a function for the time it takes the visitor to get from the cabin to the island.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q212787\">Hint<\/button><\/p>\n<div id=\"q212787\" class=\"hidden-answer\" style=\"display: none\">\n<p>The time [latex]T=T_{\\text{running}}+T_{\\text{swimming}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042369705\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042369705\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042369705\">[latex]T(x)=\\frac{x}{6}+\\frac{\\sqrt{(15-x)^2+1}}{2.5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165042376530\">A car rental company charges its customers [latex]p[\/latex] dollars per day, where [latex]60\\le p\\le 150[\/latex]. It has found that the number of cars rented per day can be modeled by the linear function [latex]n(p)=750-5p[\/latex]. How much should the company charge each customer to maximize revenue?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q13435467\">Hint<\/button><\/p>\n<div id=\"q13435467\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042321515\">[latex]R(p)=n \\times p[\/latex], where [latex]n[\/latex] is the number of cars rented and [latex]p[\/latex] is the price charged per car.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042960136\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042960136\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042960136\">The company should charge [latex]$75[\/latex] per car per day.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165043199268\">Modify the area function [latex]A[\/latex] if the rectangle is to be inscribed in the unit circle [latex]x^2+y^2=1[\/latex]. What is the domain of consideration?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q70845661\">Hint<\/button><\/p>\n<div id=\"q70845661\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042318676\">If [latex](x,y)[\/latex] is the vertex of the square that lies in the first quadrant, then the area of the square is [latex]A=(2x)(2y)=4xy[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043276112\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043276112\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043276112\">[latex]A(x)=4x\\sqrt{1-x^2}[\/latex]. The domain of consideration is [latex][0,1][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":292,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3121"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3121\/revisions"}],"predecessor-version":[{"id":3891,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3121\/revisions\/3891"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/292"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3121\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3121"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3121"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3121"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3121"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}