{"id":320,"date":"2019-07-15T22:44:10","date_gmt":"2019-07-15T22:44:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/inverse-variation\/"},"modified":"2019-07-15T22:44:10","modified_gmt":"2019-07-15T22:44:10","slug":"inverse-variation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/inverse-variation\/","title":{"raw":"Inverse and Joint Variation","rendered":"Inverse and Joint Variation"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Solve an Inverse variation problem.<\/li>\n \t<li>Write a formula for an inversely proportional relationship.<\/li>\n<\/ul>\n<\/div>\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.\n\nIf we create a table&nbsp;we observe that, as the depth increases, the water temperature decreases.\n<table summary=\"..\">\n<thead>\n<tr>\n<th>[latex]d[\/latex], 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>\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>.\n\nFor our example, the graph&nbsp;depicts 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=\\dfrac{k}{x}[\/latex] for inverse variation in this case uses [latex]k=14,000[\/latex].\n\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\">\n<div class=\"textbox\">\n<h3>A General Note: Inverse Variation<\/h3>\nIf [latex]x[\/latex] and [latex]y[\/latex]&nbsp;are related by an equation of the form\n\n[latex]y=\\dfrac{k}{{x}^{n}}[\/latex]\n\nwhere [latex]k[\/latex]&nbsp;is a nonzero constant, then we say that [latex]y[\/latex]&nbsp;<strong>varies inversely<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex]. In <strong>inversely proportional<\/strong> relationships, or <strong>inverse variations<\/strong>, there is a constant multiple [latex]k={x}^{n}y[\/latex].\n\n<\/div>\n<div class=\"textbox examples\">\n<h3>isolating the constant of variation<\/h3>\nTo isolate the constant of variation in an inverse variation, use the properties of equality to solve the equation for [latex]k[\/latex].\n\n[latex]y=\\dfrac{k}{{x}^{n}}[\/latex]\n\nIsolate [latex]k[\/latex] using algebra.\n\n[latex]yx^n=k[\/latex]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Formula for an Inversely Proportional Relationship<\/h3>\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.\n\n[reveal-answer q=\"81111\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"81111\"]\n\nRecall that multiplying speed by time gives distance. If we let [latex]t[\/latex]&nbsp;represent the drive time in hours, and [latex]v[\/latex]&nbsp;represent the velocity (speed or rate) at which the tourist drives, then [latex]vt=[\/latex]&nbsp;distance. Because the distance is fixed at 100 miles, [latex]vt=100[\/latex]. Solving this relationship for the time gives us our function.\n<p style=\"text-align: center\">[latex]\\begin{align}t\\left(v\\right)&amp;=\\dfrac{100}{v} \\\\[1mm] &amp;=100{v}^{-1} \\end{align}[\/latex]<\/p>\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.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a description of an inverse variation problem, solve for an unknown.<strong>\n<\/strong><\/h3>\n<ol>\n \t<li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\n \t<li>Determine the constant of variation. You may need to multiply [latex]y[\/latex]&nbsp;by the specified power of [latex]x[\/latex]&nbsp;to determine the constant of variation.<\/li>\n \t<li>Use the constant of variation to write an equation for the relationship.<\/li>\n \t<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>\nA quantity [latex]y[\/latex]&nbsp;varies inversely with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]&nbsp;when [latex]x=2[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 6.\n\n[reveal-answer q=\"482072\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"482072\"]\n\nThe general formula for inverse variation with a cube is [latex]y=\\dfrac{k}{{x}^{3}}[\/latex]. The constant can be found by multiplying [latex]y[\/latex]&nbsp;by the cube of [latex]x[\/latex].\n<p style=\"text-align: center\">[latex]\\begin{align}k&amp;={x}^{3}y \\\\[1mm] &amp;={2}^{3}\\cdot 25 \\\\[1mm] &amp;=200 \\end{align}[\/latex]<\/p>\nNow we use the constant to write an equation that represents this relationship.\n<p style=\"text-align: center\">[latex]\\begin{align}y&amp;=\\dfrac{k}{{x}^{3}},\\hspace{2mm}k=200 \\\\[1mm] y&amp;=\\dfrac{200}{{x}^{3}} \\end{align}[\/latex]<\/p>\nSubstitute [latex]x=6[\/latex]&nbsp;and solve for [latex]y[\/latex].\n<p style=\"text-align: center\">[latex]\\begin{align}y&amp;=\\dfrac{200}{{6}^{3}} \\\\[1mm] &amp;=\\dfrac{25}{27} \\end{align}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nThe graph of this equation is a rational function.\n\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\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nA quantity [latex]y[\/latex]&nbsp;varies inversely with the square of [latex]x[\/latex]. If [latex]y=8[\/latex]&nbsp;when [latex]x=3[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 4.\n\n[reveal-answer q=\"285259\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"285259\"]\n\n[latex]\\dfrac{9}{2}[\/latex]\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91393&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\n\n<\/div>\nThe following video presents a short lesson on inverse variation and includes more worked examples.\n\nhttps:\/\/youtu.be\/awp2vxqd-l4\n<h2>Joint Variation<\/h2>\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 [latex]c[\/latex], cost, varies jointly with the number of students, [latex]n[\/latex], and the distance, [latex]d[\/latex].\n<div class=\"textbox\">\n<h3>A General Note: Joint Variation<\/h3>\nJoint variation occurs when a variable varies directly or inversely with multiple variables.\n\nFor instance, if [latex]x[\/latex]&nbsp;varies directly with both [latex]y[\/latex]&nbsp;and [latex]z[\/latex], we have [latex]x=kyz[\/latex]. If [latex]x[\/latex]&nbsp;varies directly with [latex]y[\/latex]&nbsp;and inversely with [latex]z[\/latex], we have [latex]x=\\dfrac{ky}{z}[\/latex]. Notice that we only use one constant in a joint variation equation.\n\n<\/div>\n<div class=\"textbox examples\">\n<h3>isolating the constant of variation<\/h3>\nTo isolate the constant of variation in a joint variation, use the properties of equality to solve the equation for [latex]k[\/latex].\n\n[latex]x=kyz[\/latex]\n\nIsolate [latex]k[\/latex] using algebra.\n\n[latex]\\dfrac{x}{yz}=k[\/latex]\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Problems Involving Joint Variation<\/h3>\nA quantity [latex]x[\/latex]&nbsp;varies directly with the square of [latex]y[\/latex]&nbsp;and inversely with the cube root of [latex]z[\/latex]. If [latex]x=6[\/latex]&nbsp;when [latex]y=2[\/latex]&nbsp;and [latex]z=8[\/latex], find [latex]x[\/latex]&nbsp;when [latex]y=1[\/latex]&nbsp;and [latex]z=27[\/latex].\n\n[reveal-answer q=\"396823\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"396823\"]\n\nBegin by writing an equation to show the relationship between the variables.\n<p style=\"text-align: center\">[latex]x=\\dfrac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\nSubstitute [latex]x=6[\/latex], [latex]y=2[\/latex], and [latex]z=8[\/latex]&nbsp;to find the value of the constant [latex]k[\/latex].\n<p style=\"text-align: center\">[latex]\\begin{align}6&amp;=\\dfrac{k{2}^{2}}{\\sqrt[3]{8}} \\\\[1mm] 6&amp;=\\dfrac{4k}{2} \\\\[1mm] 3&amp;=k \\end{align}[\/latex]<\/p>\nNow we can substitute the value of the constant into the equation for the relationship.\n<p style=\"text-align: center\">[latex]x=\\dfrac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\nTo find [latex]x[\/latex]&nbsp;when [latex]y=1[\/latex]&nbsp;and [latex]z=27[\/latex], we will substitute values for [latex]y[\/latex]&nbsp;and [latex]z[\/latex]&nbsp;into our equation.\n<p style=\"text-align: center\">[latex]\\begin{align}x&amp;=\\dfrac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}} \\\\[1mm] &amp;=1 \\end{align}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[latex]x[\/latex] varies directly with the square of [latex]y[\/latex]&nbsp;and inversely with [latex]z[\/latex]. If [latex]x=40[\/latex]&nbsp;when [latex]y=4[\/latex]&nbsp;and [latex]z=2[\/latex], find [latex]x[\/latex]&nbsp;when [latex]y=10[\/latex]&nbsp;and [latex]z=25[\/latex].\n\n[reveal-answer q=\"286100\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"286100\"]\n\n[latex]x=20[\/latex]\n\n[\/hidden-answer]\n\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91394&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\n\n<\/div>\n&nbsp;\n\nThe following video provides another worked example of a joint variation problem.\n\nhttps:\/\/youtu.be\/JREPATMScbM\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/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&nbsp;we observe that, as the depth increases, the water temperature decreases.<\/p>\n<table summary=\"..\">\n<thead>\n<tr>\n<th>[latex]d[\/latex], 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&nbsp;depicts 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=\\dfrac{k}{x}[\/latex] for inverse variation in this case uses [latex]k=14,000[\/latex].<\/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\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Inverse Variation<\/h3>\n<p>If [latex]x[\/latex] and [latex]y[\/latex]&nbsp;are related by an equation of the form<\/p>\n<p>[latex]y=\\dfrac{k}{{x}^{n}}[\/latex]<\/p>\n<p>where [latex]k[\/latex]&nbsp;is a nonzero constant, then we say that [latex]y[\/latex]&nbsp;<strong>varies inversely<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex]. 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 examples\">\n<h3>isolating the constant of variation<\/h3>\n<p>To isolate the constant of variation in an inverse variation, use the properties of equality to solve the equation for [latex]k[\/latex].<\/p>\n<p>[latex]y=\\dfrac{k}{{x}^{n}}[\/latex]<\/p>\n<p>Isolate [latex]k[\/latex] using algebra.<\/p>\n<p>[latex]yx^n=k[\/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\">Show Solution<\/span><\/p>\n<div id=\"q81111\" class=\"hidden-answer\" style=\"display: none\">\n<p>Recall that multiplying speed by time gives distance. If we let [latex]t[\/latex]&nbsp;represent the drive time in hours, and [latex]v[\/latex]&nbsp;represent the velocity (speed or rate) at which the tourist drives, then [latex]vt=[\/latex]&nbsp;distance. Because the distance is fixed at 100 miles, [latex]vt=100[\/latex]. Solving this relationship for the time gives us our function.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}t\\left(v\\right)&=\\dfrac{100}{v} \\\\[1mm] &=100{v}^{-1} \\end{align}[\/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 inverse variation problem, solve for an unknown.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\n<li>Determine the constant of variation. You may need to multiply [latex]y[\/latex]&nbsp;by the specified power of [latex]x[\/latex]&nbsp;to 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 [latex]y[\/latex]&nbsp;varies inversely with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]&nbsp;when [latex]x=2[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 6.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q482072\">Show Solution<\/span><\/p>\n<div id=\"q482072\" class=\"hidden-answer\" style=\"display: none\">\n<p>The general formula for inverse variation with a cube is [latex]y=\\dfrac{k}{{x}^{3}}[\/latex]. The constant can be found by multiplying [latex]y[\/latex]&nbsp;by the cube of [latex]x[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}k&={x}^{3}y \\\\[1mm] &={2}^{3}\\cdot 25 \\\\[1mm] &=200 \\end{align}[\/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{align}y&=\\dfrac{k}{{x}^{3}},\\hspace{2mm}k=200 \\\\[1mm] y&=\\dfrac{200}{{x}^{3}} \\end{align}[\/latex]<\/p>\n<p>Substitute [latex]x=6[\/latex]&nbsp;and solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}y&=\\dfrac{200}{{6}^{3}} \\\\[1mm] &=\\dfrac{25}{27} \\end{align}[\/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\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A quantity [latex]y[\/latex]&nbsp;varies inversely with the square of [latex]x[\/latex]. If [latex]y=8[\/latex]&nbsp;when [latex]x=3[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 4.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q285259\">Show Solution<\/span><\/p>\n<div id=\"q285259\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{9}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm91393\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91393&#38;theme=oea&#38;iframe_resize_id=ohm91393&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/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 [latex]c[\/latex], cost, varies jointly with the number of students, [latex]n[\/latex], and the distance, [latex]d[\/latex].<\/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 [latex]x[\/latex]&nbsp;varies directly with both [latex]y[\/latex]&nbsp;and [latex]z[\/latex], we have [latex]x=kyz[\/latex]. If [latex]x[\/latex]&nbsp;varies directly with [latex]y[\/latex]&nbsp;and inversely with [latex]z[\/latex], we have [latex]x=\\dfrac{ky}{z}[\/latex]. Notice that we only use one constant in a joint variation equation.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>isolating the constant of variation<\/h3>\n<p>To isolate the constant of variation in a joint variation, use the properties of equality to solve the equation for [latex]k[\/latex].<\/p>\n<p>[latex]x=kyz[\/latex]<\/p>\n<p>Isolate [latex]k[\/latex] using algebra.<\/p>\n<p>[latex]\\dfrac{x}{yz}=k[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Problems Involving Joint Variation<\/h3>\n<p>A quantity [latex]x[\/latex]&nbsp;varies directly with the square of [latex]y[\/latex]&nbsp;and inversely with the cube root of [latex]z[\/latex]. If [latex]x=6[\/latex]&nbsp;when [latex]y=2[\/latex]&nbsp;and [latex]z=8[\/latex], find [latex]x[\/latex]&nbsp;when [latex]y=1[\/latex]&nbsp;and [latex]z=27[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q396823\">Show Solution<\/span><\/p>\n<div id=\"q396823\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by writing an equation to show the relationship between the variables.<\/p>\n<p style=\"text-align: center\">[latex]x=\\dfrac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\n<p>Substitute [latex]x=6[\/latex], [latex]y=2[\/latex], and [latex]z=8[\/latex]&nbsp;to find the value of the constant [latex]k[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}6&=\\dfrac{k{2}^{2}}{\\sqrt[3]{8}} \\\\[1mm] 6&=\\dfrac{4k}{2} \\\\[1mm] 3&=k \\end{align}[\/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=\\dfrac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\n<p>To find [latex]x[\/latex]&nbsp;when [latex]y=1[\/latex]&nbsp;and [latex]z=27[\/latex], we will substitute values for [latex]y[\/latex]&nbsp;and [latex]z[\/latex]&nbsp;into our equation.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}x&=\\dfrac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}} \\\\[1mm] &=1 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>[latex]x[\/latex] varies directly with the square of [latex]y[\/latex]&nbsp;and inversely with [latex]z[\/latex]. If [latex]x=40[\/latex]&nbsp;when [latex]y=4[\/latex]&nbsp;and [latex]z=2[\/latex], find [latex]x[\/latex]&nbsp;when [latex]y=10[\/latex]&nbsp;and [latex]z=25[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q286100\">Show Solution<\/span><\/p>\n<div id=\"q286100\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=20[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm91394\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91394&#38;theme=oea&#38;iframe_resize_id=ohm91394&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/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-320\">\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":17533,"menu_order":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 91393,91394\",\"author\":\"Jenck,Michael (for Lumen Learning)\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Inverse Variation\",\"author\":\"James Sousa (Mathispower4u.com) 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