{"id":1951,"date":"2016-11-02T22:46:15","date_gmt":"2016-11-02T22:46:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1951"},"modified":"2017-04-04T19:27:37","modified_gmt":"2017-04-04T19:27:37","slug":"inverse-variation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/inverse-variation\/","title":{"raw":"Inverse and Joint Variation","rendered":"Inverse and Joint Variation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Solve an Inverse variation problem<\/li>\r\n \t<li>Write a formula for an inversely proportional relationship<\/li>\r\n<\/ul>\r\n<\/div>\r\nWater temperature in an ocean varies inversely to the water\u2019s depth. Between the depths of 250 feet and 500 feet, the formula [latex]T=\\frac{14,000}{d}[\/latex] gives us the temperature in degrees Fahrenheit at a depth in feet below Earth\u2019s surface. Consider the Atlantic Ocean, which covers 22% of Earth\u2019s surface. At a certain location, at the depth of 500 feet, the temperature may be 28\u00b0F.\r\n\r\nIf we create a table\u00a0we observe that, as the depth increases, the water temperature decreases.\r\n<table summary=\"..\">\r\n<thead>\r\n<tr>\r\n<th><em>d<\/em>, depth<\/th>\r\n<th>[latex]T=\\frac{\\text{14,000}}{d}[\/latex]<\/th>\r\n<th>Interpretation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>500 ft<\/td>\r\n<td>[latex]\\frac{14,000}{500}=28[\/latex]<\/td>\r\n<td>At a depth of 500 ft, the water temperature is 28\u00b0 F.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>350 ft<\/td>\r\n<td>[latex]\\frac{14,000}{350}=40[\/latex]<\/td>\r\n<td>At a depth of 350 ft, the water temperature is 40\u00b0 F.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>250 ft<\/td>\r\n<td>[latex]\\frac{14,000}{250}=56[\/latex]<\/td>\r\n<td>At a depth of 250 ft, the water temperature is 56\u00b0 F.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be <strong>inversely proportional<\/strong> and each term <strong>varies inversely<\/strong> with the other. Inversely proportional relationships are also called <strong>inverse variations<\/strong>.\r\n\r\nFor our example, the graph\u00a0depicts the <strong>inverse variation<\/strong>. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula [latex]y=\\frac{k}{x}[\/latex] for inverse variation in this case uses <em>k\u00a0<\/em>= 14,000.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222955\/CNX_Precalc_Figure_03_09_0032.jpg\" alt=\"Graph of y=(14000)\/x where the horizontal axis is labeled,\" width=\"487\" height=\"309\" data-media-type=\"image\/jpg\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Inverse Variation<\/h3>\r\nIf <em>x<\/em>\u00a0and <em>y<\/em>\u00a0are related by an equation of the form\r\n\r\n[latex]y=\\frac{k}{{x}^{n}}[\/latex]\r\n\r\nwhere <em>k<\/em>\u00a0is a nonzero constant, then we say that <em>y<\/em>\u00a0<strong>varies inversely<\/strong> with the <em>n<\/em>th power of <em>x<\/em>. In <strong>inversely proportional<\/strong> relationships, or <strong>inverse variations<\/strong>, there is a constant multiple [latex]k={x}^{n}y[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing a Formula for an Inversely Proportional Relationship<\/h3>\r\nA tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.\r\n\r\n[reveal-answer q=\"81111\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"81111\"]\r\nRecall that multiplying speed by time gives distance. If we let <em>t<\/em>\u00a0represent the drive time in hours, and <em>v<\/em>\u00a0represent the velocity (speed or rate) at which the tourist drives, then <em>vt\u00a0<\/em>= distance. Because the distance is fixed at 100 miles, <em>vt\u00a0<\/em>= 100. Solving this relationship for the time gives us our function.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}t\\left(v\\right)=\\frac{100}{v}\\hfill \\\\ \\text{ }=100{v}^{-1}\\hfill \\end{array}[\/latex]<\/p>\r\nWe can see that the constant of variation is 100 and, although we can write the relationship using the negative exponent, it is more common to see it written as a fraction.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a description of an indirect variation problem, solve for an unknown.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Identify the input, <em>x<\/em>, and the output, <em>y<\/em>.<\/li>\r\n \t<li>Determine the constant of variation. You may need to multiply <em>y<\/em>\u00a0by the specified power of <em>x<\/em>\u00a0to determine the constant of variation.<\/li>\r\n \t<li>Use the constant of variation to write an equation for the relationship.<\/li>\r\n \t<li>Substitute known values into the equation to find the unknown.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving an Inverse Variation Problem<\/h3>\r\nA quantity <em>y<\/em>\u00a0varies inversely with the cube of <em>x<\/em>. If <em>y\u00a0<\/em>= 25 when <em>x\u00a0<\/em>= 2, find <em>y<\/em>\u00a0when <em>x<\/em>\u00a0is 6.\r\n\r\n[reveal-answer q=\"482072\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"482072\"]\r\nThe general formula for inverse variation with a cube is [latex]y=\\frac{k}{{x}^{3}}[\/latex]. The constant can be found by multiplying <em>y<\/em>\u00a0by the cube of <em>x<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}k={x}^{3}y\\hfill \\\\ \\text{ }={2}^{3}\\cdot 25\\hfill \\\\ \\text{ }=200\\hfill \\end{array}[\/latex]<\/p>\r\nNow we use the constant to write an equation that represents this relationship.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}y=\\frac{k}{{x}^{3}},k=200\\hfill \\\\ y=\\frac{200}{{x}^{3}}\\hfill \\end{array}[\/latex]<\/p>\r\nSubstitute <em>x\u00a0<\/em>= 6 and solve for <i>y<\/i>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}y=\\frac{200}{{6}^{3}}\\hfill \\\\ \\text{ }=\\frac{25}{27}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThe graph of this equation is a rational function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222957\/CNX_Precalc_Figure_03_09_0042.jpg\" alt=\"Graph of y=25\/(x^3) with the labeled points (2, 25) and (6, 25\/27).\" width=\"488\" height=\"292\" data-media-type=\"image\/jpg\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA quantity <em>y<\/em>\u00a0varies inversely with the square of <em>x<\/em>. If <em>y\u00a0<\/em>= 8 when <em>x\u00a0<\/em>= 3, find <em>y<\/em>\u00a0when <em>x<\/em>\u00a0is 4.\r\n\r\n[reveal-answer q=\"285259\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"285259\"] [latex]\\frac{9}{2}[\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=91393&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"400\">\r\n<\/iframe>\r\n\r\n<\/div>\r\nThe following video presents a short lesson on inverse variation and includes more worked examples.\r\n\r\nhttps:\/\/youtu.be\/awp2vxqd-l4\r\n<h2>Joint Variation<\/h2>\r\nMany situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called <strong>joint variation<\/strong>. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable <em>c<\/em>, cost, varies jointly with the number of students, <em>n<\/em>, and the distance, <em>d<\/em>.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Joint Variation<\/h3>\r\nJoint variation occurs when a variable varies directly or inversely with multiple variables.\r\n\r\nFor instance, if <em>x<\/em>\u00a0varies directly with both <em>y<\/em>\u00a0and <em>z<\/em>, we have <em>x\u00a0<\/em>= <em>kyz<\/em>. If <em>x<\/em>\u00a0varies directly with <em>y<\/em>\u00a0and inversely with <em>z<\/em>, we have [latex]x=\\frac{ky}{z}[\/latex]. Notice that we only use one constant in a joint variation equation.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Example: Solving Problems Involving Joint Variation<\/h3>\r\nA quantity <em>x<\/em>\u00a0varies directly with the square of <em>y<\/em>\u00a0and inversely with the cube root of <em>z<\/em>. If <em>x\u00a0<\/em>= 6 when <em>y\u00a0<\/em>= 2 and <em>z\u00a0<\/em>= 8, find <em>x<\/em>\u00a0when <em>y\u00a0<\/em>= 1 and <em>z\u00a0<\/em>= 27.\r\n\r\n[reveal-answer q=\"396823\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"396823\"]\r\nBegin by writing an equation to show the relationship between the variables.\r\n<p style=\"text-align: center;\">[latex]x=\\frac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\r\nSubstitute <em>x\u00a0<\/em>= 6, <em>y\u00a0<\/em>= 2, and <em>z\u00a0<\/em>= 8 to find the value of the constant <em>k<\/em>.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}6=\\frac{k{2}^{2}}{\\sqrt[3]{8}}\\hfill \\\\ 6=\\frac{4k}{2}\\hfill \\\\ 3=k\\hfill \\end{array}[\/latex]<\/p>\r\nNow we can substitute the value of the constant into the equation for the relationship.\r\n<p style=\"text-align: center;\">[latex]x=\\frac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\r\nTo find <em>x<\/em>\u00a0when <em>y\u00a0<\/em>= 1 and <em>z\u00a0<\/em>= 27, we will substitute values for <em>y<\/em>\u00a0and <em>z<\/em>\u00a0into our equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x=\\frac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}}\\hfill \\\\ \\text{ }=1\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<em>x<\/em>\u00a0varies directly with the square of <em>y<\/em>\u00a0and inversely with <em>z<\/em>. If <em>x\u00a0<\/em>= 40 when <em>y\u00a0<\/em>= 4 and <em>z\u00a0<\/em>= 2, find <em>x<\/em>\u00a0when <em>y\u00a0<\/em>= 10 and <em>z\u00a0<\/em>= 25.\r\n\r\n[reveal-answer q=\"286100\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"286100\"][latex]x=20[\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=91394&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"400\">\r\n<\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe following video provides another worked example of a joint variation problem.\r\n\r\nhttps:\/\/youtu.be\/JREPATMScbM","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Solve an Inverse variation problem<\/li>\n<li>Write a formula for an inversely proportional relationship<\/li>\n<\/ul>\n<\/div>\n<p>Water temperature in an ocean varies inversely to the water\u2019s depth. Between the depths of 250 feet and 500 feet, the formula [latex]T=\\frac{14,000}{d}[\/latex] gives us the temperature in degrees Fahrenheit at a depth in feet below Earth\u2019s surface. Consider the Atlantic Ocean, which covers 22% of Earth\u2019s surface. At a certain location, at the depth of 500 feet, the temperature may be 28\u00b0F.<\/p>\n<p>If we create a table\u00a0we observe that, as the depth increases, the water temperature decreases.<\/p>\n<table summary=\"..\">\n<thead>\n<tr>\n<th><em>d<\/em>, depth<\/th>\n<th>[latex]T=\\frac{\\text{14,000}}{d}[\/latex]<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>500 ft<\/td>\n<td>[latex]\\frac{14,000}{500}=28[\/latex]<\/td>\n<td>At a depth of 500 ft, the water temperature is 28\u00b0 F.<\/td>\n<\/tr>\n<tr>\n<td>350 ft<\/td>\n<td>[latex]\\frac{14,000}{350}=40[\/latex]<\/td>\n<td>At a depth of 350 ft, the water temperature is 40\u00b0 F.<\/td>\n<\/tr>\n<tr>\n<td>250 ft<\/td>\n<td>[latex]\\frac{14,000}{250}=56[\/latex]<\/td>\n<td>At a depth of 250 ft, the water temperature is 56\u00b0 F.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be <strong>inversely proportional<\/strong> and each term <strong>varies inversely<\/strong> with the other. Inversely proportional relationships are also called <strong>inverse variations<\/strong>.<\/p>\n<p>For our example, the graph\u00a0depicts the <strong>inverse variation<\/strong>. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula [latex]y=\\frac{k}{x}[\/latex] for inverse variation in this case uses <em>k\u00a0<\/em>= 14,000.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222955\/CNX_Precalc_Figure_03_09_0032.jpg\" alt=\"Graph of y=(14000)\/x where the horizontal axis is labeled,\" width=\"487\" height=\"309\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Inverse Variation<\/h3>\n<p>If <em>x<\/em>\u00a0and <em>y<\/em>\u00a0are related by an equation of the form<\/p>\n<p>[latex]y=\\frac{k}{{x}^{n}}[\/latex]<\/p>\n<p>where <em>k<\/em>\u00a0is a nonzero constant, then we say that <em>y<\/em>\u00a0<strong>varies inversely<\/strong> with the <em>n<\/em>th power of <em>x<\/em>. In <strong>inversely proportional<\/strong> relationships, or <strong>inverse variations<\/strong>, there is a constant multiple [latex]k={x}^{n}y[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Formula for an Inversely Proportional Relationship<\/h3>\n<p>A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q81111\">Solution<\/span><\/p>\n<div id=\"q81111\" class=\"hidden-answer\" style=\"display: none\">\nRecall that multiplying speed by time gives distance. If we let <em>t<\/em>\u00a0represent the drive time in hours, and <em>v<\/em>\u00a0represent the velocity (speed or rate) at which the tourist drives, then <em>vt\u00a0<\/em>= distance. Because the distance is fixed at 100 miles, <em>vt\u00a0<\/em>= 100. Solving this relationship for the time gives us our function.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}t\\left(v\\right)=\\frac{100}{v}\\hfill \\\\ \\text{ }=100{v}^{-1}\\hfill \\end{array}[\/latex]<\/p>\n<p>We can see that the constant of variation is 100 and, although we can write the relationship using the negative exponent, it is more common to see it written as a fraction.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a description of an indirect variation problem, solve for an unknown.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Identify the input, <em>x<\/em>, and the output, <em>y<\/em>.<\/li>\n<li>Determine the constant of variation. You may need to multiply <em>y<\/em>\u00a0by the specified power of <em>x<\/em>\u00a0to determine the constant of variation.<\/li>\n<li>Use the constant of variation to write an equation for the relationship.<\/li>\n<li>Substitute known values into the equation to find the unknown.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Inverse Variation Problem<\/h3>\n<p>A quantity <em>y<\/em>\u00a0varies inversely with the cube of <em>x<\/em>. If <em>y\u00a0<\/em>= 25 when <em>x\u00a0<\/em>= 2, find <em>y<\/em>\u00a0when <em>x<\/em>\u00a0is 6.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q482072\">Solution<\/span><\/p>\n<div id=\"q482072\" class=\"hidden-answer\" style=\"display: none\">\nThe general formula for inverse variation with a cube is [latex]y=\\frac{k}{{x}^{3}}[\/latex]. The constant can be found by multiplying <em>y<\/em>\u00a0by the cube of <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}k={x}^{3}y\\hfill \\\\ \\text{ }={2}^{3}\\cdot 25\\hfill \\\\ \\text{ }=200\\hfill \\end{array}[\/latex]<\/p>\n<p>Now we use the constant to write an equation that represents this relationship.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}y=\\frac{k}{{x}^{3}},k=200\\hfill \\\\ y=\\frac{200}{{x}^{3}}\\hfill \\end{array}[\/latex]<\/p>\n<p>Substitute <em>x\u00a0<\/em>= 6 and solve for <i>y<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}y=\\frac{200}{{6}^{3}}\\hfill \\\\ \\text{ }=\\frac{25}{27}\\hfill \\end{array}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph of this equation is a rational function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222957\/CNX_Precalc_Figure_03_09_0042.jpg\" alt=\"Graph of y=25\/(x^3) with the labeled points (2, 25) and (6, 25\/27).\" width=\"488\" height=\"292\" data-media-type=\"image\/jpg\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A quantity <em>y<\/em>\u00a0varies inversely with the square of <em>x<\/em>. If <em>y\u00a0<\/em>= 8 when <em>x\u00a0<\/em>= 3, find <em>y<\/em>\u00a0when <em>x<\/em>\u00a0is 4.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q285259\">Solution<\/span><\/p>\n<div id=\"q285259\" class=\"hidden-answer\" style=\"display: none\"> [latex]\\frac{9}{2}[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=91393&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"400\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<p>The following video presents a short lesson on inverse variation and includes more worked examples.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Inverse Variation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/awp2vxqd-l4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Joint Variation<\/h2>\n<p>Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called <strong>joint variation<\/strong>. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable <em>c<\/em>, cost, varies jointly with the number of students, <em>n<\/em>, and the distance, <em>d<\/em>.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Joint Variation<\/h3>\n<p>Joint variation occurs when a variable varies directly or inversely with multiple variables.<\/p>\n<p>For instance, if <em>x<\/em>\u00a0varies directly with both <em>y<\/em>\u00a0and <em>z<\/em>, we have <em>x\u00a0<\/em>= <em>kyz<\/em>. If <em>x<\/em>\u00a0varies directly with <em>y<\/em>\u00a0and inversely with <em>z<\/em>, we have [latex]x=\\frac{ky}{z}[\/latex]. Notice that we only use one constant in a joint variation equation.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Example: Solving Problems Involving Joint Variation<\/h3>\n<p>A quantity <em>x<\/em>\u00a0varies directly with the square of <em>y<\/em>\u00a0and inversely with the cube root of <em>z<\/em>. If <em>x\u00a0<\/em>= 6 when <em>y\u00a0<\/em>= 2 and <em>z\u00a0<\/em>= 8, find <em>x<\/em>\u00a0when <em>y\u00a0<\/em>= 1 and <em>z\u00a0<\/em>= 27.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q396823\">Solution<\/span><\/p>\n<div id=\"q396823\" class=\"hidden-answer\" style=\"display: none\">\nBegin by writing an equation to show the relationship between the variables.<\/p>\n<p style=\"text-align: center;\">[latex]x=\\frac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\n<p>Substitute <em>x\u00a0<\/em>= 6, <em>y\u00a0<\/em>= 2, and <em>z\u00a0<\/em>= 8 to find the value of the constant <em>k<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}6=\\frac{k{2}^{2}}{\\sqrt[3]{8}}\\hfill \\\\ 6=\\frac{4k}{2}\\hfill \\\\ 3=k\\hfill \\end{array}[\/latex]<\/p>\n<p>Now we can substitute the value of the constant into the equation for the relationship.<\/p>\n<p style=\"text-align: center;\">[latex]x=\\frac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\n<p>To find <em>x<\/em>\u00a0when <em>y\u00a0<\/em>= 1 and <em>z\u00a0<\/em>= 27, we will substitute values for <em>y<\/em>\u00a0and <em>z<\/em>\u00a0into our equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x=\\frac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}}\\hfill \\\\ \\text{ }=1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><em>x<\/em>\u00a0varies directly with the square of <em>y<\/em>\u00a0and inversely with <em>z<\/em>. If <em>x\u00a0<\/em>= 40 when <em>y\u00a0<\/em>= 4 and <em>z\u00a0<\/em>= 2, find <em>x<\/em>\u00a0when <em>y\u00a0<\/em>= 10 and <em>z\u00a0<\/em>= 25.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q286100\">Solution<\/span><\/p>\n<div id=\"q286100\" class=\"hidden-answer\" style=\"display: none\">[latex]x=20[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=91394&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"400\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The following video provides another worked example of a joint variation problem.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Joint Variation: Determine the Variation Constant (Volume of a Cone)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/JREPATMScbM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1951\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 91393,91394. <strong>Authored by<\/strong>: Jenck,Michael (for Lumen Learning). <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Inverse Variation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/awp2vxqd-l4\">https:\/\/youtu.be\/awp2vxqd-l4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Joint Variation: Determine the Variation Constant (Volume of a Cone). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Provided by<\/strong>: Joint Variation: Determine the Variation Constant (Volume of a Cone). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/JREPATMScbM\">https:\/\/youtu.be\/JREPATMScbM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 91393,91394\",\"author\":\"Jenck,Michael (for Lumen 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