{"id":1421,"date":"2025-07-25T01:12:49","date_gmt":"2025-07-25T01:12:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1421"},"modified":"2026-03-18T06:05:28","modified_gmt":"2026-03-18T06:05:28","slug":"modeling-using-variation-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/modeling-using-variation-fresh-take\/","title":{"raw":"Modeling Using Variation: Fresh Take","rendered":"Modeling Using Variation: Fresh Take"},"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>Direct Variation<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Direct Variation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A relationship where one variable is a constant multiple of another<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Expressed as [latex]y = kx^n[\/latex], where [latex]k[\/latex] is the constant of variation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Characteristics:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">As one variable increases, the other increases proportionally<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The ratio between variables remains constant: [latex]k = \\frac{y}{x^n}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Direct Variation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Linear ([latex]n = 1[\/latex]): [latex]y = kx[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadratic ([latex]n = 2[\/latex]): [latex]y = kx^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Cubic ([latex]n = 3[\/latex]): [latex]y = kx^3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">And so on for higher powers<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphical Representation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">All direct variation graphs pass through the origin [latex](0, 0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Shape depends on the power n (linear, parabola, cubic, etc.)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used in real-world situations where quantities vary consistently<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Examples: sales commissions, simple interest, distance-time relationships<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">The quantity [latex]y[\/latex]\u00a0varies directly with the square of [latex]y[\/latex]. If [latex]y=24[\/latex]\u00a0when [latex]x=3[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 4.[reveal-answer q=\"536994\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"536994\"][latex]\\dfrac{128}{3}[\/latex][\/hidden-answer]<\/section>Watch this video to see a quick lesson in direct variation. \u00a0You will see more worked examples.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ffhbdaeh-plFOq4JaEyI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/plFOq4JaEyI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ffhbdaeh-plFOq4JaEyI\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=6454977&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ffhbdaeh-plFOq4JaEyI&amp;vembed=0&amp;video_id=plFOq4JaEyI&amp;video_target=tpm-plugin-ffhbdaeh-plFOq4JaEyI\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Direct+Variation+Applications_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDirect Variation Applications\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Inverse Variation<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Inverse Variation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A relationship where one variable decreases as the other increases<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Expressed as [latex]y = \\frac{k}{x^n}[\/latex], where [latex]k[\/latex] is the constant of variation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Characteristics:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">As one variable increases, the other decreases proportionally<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The product of the variables remains constant: [latex]k = x^n \\cdot y[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Inverse Variation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Simple inverse ([latex]n = 1[\/latex]): [latex]y = \\frac{k}{x}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inverse square ([latex]n = 2[\/latex]): [latex]y = \\frac{k}{x^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inverse cube ([latex]n = 3[\/latex]): [latex]y = \\frac{k}{x^3}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">And so on for higher powers<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Graphical Representation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Hyperbola for simple inverse variation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Asymptotic to both axes (never touches x or y axis)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Not defined when [latex]x = 0[\/latex] (division by zero)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used in physics (Boyle's Law, gravitational force)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Economics (supply and demand)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Engineering (gear ratios)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A quantity [latex]y[\/latex]\u00a0varies inversely with the square of [latex]x[\/latex]. If [latex]y=8[\/latex]\u00a0when [latex]x=3[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is [latex]4[\/latex].[reveal-answer q=\"285259\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"285259\"][latex]\\dfrac{9}{2}[\/latex][\/hidden-answer]<\/section>The following video presents a short lesson on inverse variation and includes more worked examples.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dcgfbheg-awp2vxqd-l4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/awp2vxqd-l4?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dcgfbheg-awp2vxqd-l4\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850241&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dcgfbheg-awp2vxqd-l4&amp;vembed=0&amp;video_id=awp2vxqd-l4&amp;video_target=tpm-plugin-dcgfbheg-awp2vxqd-l4\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Inverse+Variation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cInverse Variation\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Joint Variation<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Joint Variation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A relationship where a variable depends on two or more other variables<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can involve both direct and inverse variations simultaneously<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Forms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Direct joint variation: [latex]x = kyz[\/latex] ([latex]x[\/latex] varies directly with both [latex]y[\/latex] and [latex]z[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Mixed joint variation: [latex]x = \\frac{ky}{z}[\/latex] ([latex]x[\/latex] varies directly with [latex]y[\/latex] and inversely with [latex]z[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Characteristics:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Only one constant (k) is used in the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can involve different powers of variables<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Formula: [latex]x = k \\frac{y^a z^b}{w^c}[\/latex]\r\nWhere [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers (positive for direct variation, negative for inverse variation)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Physics (force, pressure, work)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Engineering (stress and strain relationships)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Economics (production functions)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">[latex]x[\/latex] varies directly with the square of [latex]y[\/latex]\u00a0and inversely with [latex]z[\/latex]. If [latex]x=40[\/latex]\u00a0when [latex]y=4[\/latex]\u00a0and [latex]z=2[\/latex], find [latex]x[\/latex]\u00a0when [latex]y=10[\/latex]\u00a0and [latex]z=25[\/latex].[reveal-answer q=\"286100\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"286100\"][latex]x=20[\/latex][\/hidden-answer]<\/section>The following video provides another worked example of a joint variation problem.\r\n\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ahgecfdb-JREPATMScbM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/JREPATMScbM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ahgecfdb-JREPATMScbM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850240&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ahgecfdb-JREPATMScbM&amp;vembed=0&amp;video_id=JREPATMScbM&amp;video_target=tpm-plugin-ahgecfdb-JREPATMScbM\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Joint+Variation+-+Determine+the+Variation+Constant+(Volume+of+a+Cone)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cJoint Variation: Determine the Variation Constant (Volume of a Cone)\u201d here (opens in new window).<\/a>\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>Direct Variation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Direct Variation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A relationship where one variable is a constant multiple of another<\/li>\n<li class=\"whitespace-normal break-words\">Expressed as [latex]y = kx^n[\/latex], where [latex]k[\/latex] is the constant of variation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">As one variable increases, the other increases proportionally<\/li>\n<li class=\"whitespace-normal break-words\">The ratio between variables remains constant: [latex]k = \\frac{y}{x^n}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Direct Variation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Linear ([latex]n = 1[\/latex]): [latex]y = kx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Quadratic ([latex]n = 2[\/latex]): [latex]y = kx^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Cubic ([latex]n = 3[\/latex]): [latex]y = kx^3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">And so on for higher powers<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphical Representation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">All direct variation graphs pass through the origin [latex](0, 0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Shape depends on the power n (linear, parabola, cubic, etc.)<\/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\">Used in real-world situations where quantities vary consistently<\/li>\n<li class=\"whitespace-normal break-words\">Examples: sales commissions, simple interest, distance-time relationships<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">The quantity [latex]y[\/latex]\u00a0varies directly with the square of [latex]y[\/latex]. If [latex]y=24[\/latex]\u00a0when [latex]x=3[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 4.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q536994\">Show Solution<\/button><\/p>\n<div id=\"q536994\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{128}{3}[\/latex]<\/div>\n<\/div>\n<\/section>\n<p>Watch this video to see a quick lesson in direct variation. \u00a0You will see more worked examples.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ffhbdaeh-plFOq4JaEyI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/plFOq4JaEyI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ffhbdaeh-plFOq4JaEyI\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=6454977&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ffhbdaeh-plFOq4JaEyI&amp;vembed=0&amp;video_id=plFOq4JaEyI&amp;video_target=tpm-plugin-ffhbdaeh-plFOq4JaEyI\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Direct+Variation+Applications_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDirect Variation Applications\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Inverse Variation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Inverse Variation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A relationship where one variable decreases as the other increases<\/li>\n<li class=\"whitespace-normal break-words\">Expressed as [latex]y = \\frac{k}{x^n}[\/latex], where [latex]k[\/latex] is the constant of variation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">As one variable increases, the other decreases proportionally<\/li>\n<li class=\"whitespace-normal break-words\">The product of the variables remains constant: [latex]k = x^n \\cdot y[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Inverse Variation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simple inverse ([latex]n = 1[\/latex]): [latex]y = \\frac{k}{x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Inverse square ([latex]n = 2[\/latex]): [latex]y = \\frac{k}{x^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Inverse cube ([latex]n = 3[\/latex]): [latex]y = \\frac{k}{x^3}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">And so on for higher powers<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphical Representation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Hyperbola for simple inverse variation<\/li>\n<li class=\"whitespace-normal break-words\">Asymptotic to both axes (never touches x or y axis)<\/li>\n<li class=\"whitespace-normal break-words\">Not defined when [latex]x = 0[\/latex] (division by zero)<\/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\">Used in physics (Boyle&#8217;s Law, gravitational force)<\/li>\n<li class=\"whitespace-normal break-words\">Economics (supply and demand)<\/li>\n<li class=\"whitespace-normal break-words\">Engineering (gear ratios)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A quantity [latex]y[\/latex]\u00a0varies inversely with the square of [latex]x[\/latex]. If [latex]y=8[\/latex]\u00a0when [latex]x=3[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is [latex]4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q285259\">Show Solution<\/button><\/p>\n<div id=\"q285259\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{9}{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<p>The following video presents a short lesson on inverse variation and includes more worked examples.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dcgfbheg-awp2vxqd-l4\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/awp2vxqd-l4?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dcgfbheg-awp2vxqd-l4\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850241&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dcgfbheg-awp2vxqd-l4&amp;vembed=0&amp;video_id=awp2vxqd-l4&amp;video_target=tpm-plugin-dcgfbheg-awp2vxqd-l4\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Inverse+Variation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cInverse Variation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Joint Variation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Joint Variation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A relationship where a variable depends on two or more other variables<\/li>\n<li class=\"whitespace-normal break-words\">Can involve both direct and inverse variations simultaneously<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Forms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Direct joint variation: [latex]x = kyz[\/latex] ([latex]x[\/latex] varies directly with both [latex]y[\/latex] and [latex]z[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Mixed joint variation: [latex]x = \\frac{ky}{z}[\/latex] ([latex]x[\/latex] varies directly with [latex]y[\/latex] and inversely with [latex]z[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Characteristics:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Only one constant (k) is used in the equation<\/li>\n<li class=\"whitespace-normal break-words\">Can involve different powers of variables<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Formula: [latex]x = k \\frac{y^a z^b}{w^c}[\/latex]<br \/>\nWhere [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers (positive for direct variation, negative for inverse variation)<\/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\">Physics (force, pressure, work)<\/li>\n<li class=\"whitespace-normal break-words\">Engineering (stress and strain relationships)<\/li>\n<li class=\"whitespace-normal break-words\">Economics (production functions)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">[latex]x[\/latex] varies directly with the square of [latex]y[\/latex]\u00a0and inversely with [latex]z[\/latex]. If [latex]x=40[\/latex]\u00a0when [latex]y=4[\/latex]\u00a0and [latex]z=2[\/latex], find [latex]x[\/latex]\u00a0when [latex]y=10[\/latex]\u00a0and [latex]z=25[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q286100\">Show Solution<\/button><\/p>\n<div id=\"q286100\" class=\"hidden-answer\" style=\"display: none\">[latex]x=20[\/latex]<\/div>\n<\/div>\n<\/section>\n<p>The following video provides another worked example of a joint variation problem.<\/p>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ahgecfdb-JREPATMScbM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/JREPATMScbM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ahgecfdb-JREPATMScbM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12850240&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ahgecfdb-JREPATMScbM&amp;vembed=0&amp;video_id=JREPATMScbM&amp;video_target=tpm-plugin-ahgecfdb-JREPATMScbM\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Joint+Variation+-+Determine+the+Variation+Constant+(Volume+of+a+Cone)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cJoint Variation: Determine the Variation Constant (Volume of a Cone)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":24,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Direct Variation Applications\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/plFOq4JaEyI\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Inverse Variation\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/awp2vxqd-l4\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Joint Variation: Determine the Variation Constant (Volume of a Cone)\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/JREPATMScbM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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