{"id":13901,"date":"2018-08-24T22:47:28","date_gmt":"2018-08-24T22:47:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalcone\/?post_type=chapter&#038;p=13901"},"modified":"2025-02-05T05:19:19","modified_gmt":"2025-02-05T05:19:19","slug":"modeling-using-variation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/modeling-using-variation\/","title":{"raw":"Modeling Using Variation","rendered":"Modeling Using Variation"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\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<\/div>\r\n<p id=\"fs-id1165135356540\">A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.<\/p>\r\n\r\n<h2>Solve direct variation problems<\/h2>\r\n<p id=\"fs-id1165137823230\">In the example above, Nicole\u2019s earnings can be found by multiplying her sales by her commission. The formula <em>e<\/em> = 0.16<em>s<\/em> tells us her earnings, <em>e<\/em>, come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.<\/p>\r\n\r\n<table id=\"Table_03_09_01\" summary=\"..\">\r\n<thead>\r\n<tr>\r\n<th><em>s<\/em>, sales prices<\/th>\r\n<th><em>e<\/em> = 0.16<em>s<\/em><\/th>\r\n<th>Interpretation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>$4,600<\/td>\r\n<td><em>e\u00a0<\/em>= 0.16(4,600) = 736<\/td>\r\n<td>A sale of a $4,600 vehicle results in $736 earnings.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>$9,200<\/td>\r\n<td><em>e\u00a0<\/em>= 0.16(9,200) = 1,472<\/td>\r\n<td>A sale of a $9,200 vehicle results in $1472 earnings.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>$18,400<\/td>\r\n<td><em>e\u00a0<\/em>= 0.16(18,400) = 2,944<\/td>\r\n<td>A sale of a $18,400 vehicle results in $2944 earnings.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135188294\">Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called <strong>direct variation<\/strong>. Each variable in this type of relationship <strong>varies directly <\/strong>with the other.<\/p>\r\n<p id=\"fs-id1165137937533\">The graph below\u00a0represents the data for Nicole\u2019s potential earnings. We say that earnings vary directly with the sales price of the car. The formula [latex]y=k{x}^{n}[\/latex] is used for direct variation. The value <em>k<\/em>\u00a0is a nonzero constant greater than zero and is called the <strong>constant of variation<\/strong>. In this case, <em>k\u00a0<\/em>= 0.16 and <em>n\u00a0<\/em>= 1.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010805\/CNX_Precalc_Figure_03_09_0012.jpg\" alt=\"Graph of y=(0.16)x where the horizontal axis is labeled, \" width=\"487\" height=\"459\" \/> <b>Figure 1<\/b>[\/caption]\r\n\r\n<div id=\"fs-id1165137730075\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Direct Variation<\/h3>\r\n<p id=\"fs-id1165137827458\">If <em>x <\/em>and <em>y<\/em>\u00a0are related by an equation of the form<\/p>\r\n\r\n<div id=\"fs-id1165135437156\" class=\"equation\" style=\"text-align: center;\">[latex]y=k{x}^{n}[\/latex]<\/div>\r\n<p id=\"fs-id1165133155266\">then we say that the relationship is <strong>direct variation<\/strong> and <em>y<\/em>\u00a0<strong>varies directly<\/strong> with the <em>n<\/em>th power of <em>x<\/em>. In direct variation relationships, there is a nonzero constant ratio [latex]k=\\frac{y}{{x}^{n}}[\/latex], where <em>k<\/em>\u00a0is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137550958\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137723932\">How To: Given a description of a direct variation problem, solve for an unknown.<strong>\r\n<\/strong><\/h3>\r\n<ol id=\"fs-id1165137724401\">\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 divide <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 id=\"Example_03_09_01\" class=\"example\">\r\n<div id=\"fs-id1165137676066\" class=\"exercise\">\r\n<div id=\"fs-id1165137434564\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Solving a Direct Variation Problem<\/h3>\r\n<p id=\"fs-id1165137849016\">The quantity <em>y<\/em>\u00a0varies directly 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>\r\n[reveal-answer q=\"467589\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"467589\"]\r\n<p id=\"fs-id1165137659713\">The general formula for direct variation with a cube is [latex]y=k{x}^{3}[\/latex]. The constant can be found by dividing <em>y<\/em>\u00a0by the cube of <em>x<\/em>.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align} k&amp;=\\frac{y}{{x}^{3}} \\\\[1mm] &amp;=\\frac{25}{{2}^{3}}\\\\[1mm] &amp;=\\frac{25}{8}\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137628102\">Now use the constant to write an equation that represents this relationship.<\/p>\r\n<p style=\"text-align: center;\">[latex]y=\\frac{25}{8}{x}^{3}[\/latex]<\/p>\r\n<p id=\"fs-id1165135432964\">Substitute <em>x<\/em> = 6 and solve for <em>y<\/em>.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\frac{25}{8}{\\left(6\\right)}^{3} \\\\[1mm] &amp;=675 \\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165134557390\">The graph of this equation is a simple cubic, as shown below.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010805\/CNX_Precalc_Figure_03_09_0022.jpg\" alt=\"Graph of y=25\/8(x^3) with the labeled points (2, 25) and (6, 675).\" width=\"487\" height=\"367\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135533140\" class=\"commentary\">\r\n<div class=\"mceTemp\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137736204\" class=\"note precalculus qa textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<p id=\"eip-id1165137772190\"><strong>Do the graphs of all direct variation equations look like Example 1?<\/strong><\/p>\r\n<p id=\"fs-id1165137596402\"><em>No. Direct variation equations are power functions\u2014they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through (0, 0).<\/em><\/p>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135160334\">The quantity <em>y<\/em>\u00a0varies directly with the square of <em>x<\/em>. If <em>y\u00a0<\/em>= 24 when <em>x\u00a0<\/em>= 3, find <em>y<\/em>\u00a0when <em>x<\/em>\u00a0is 4.<\/p>\r\n[reveal-answer q=\"272502\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"272502\"]\r\n\r\n[latex]\\frac{128}{3}[\/latex]\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\n[ohm_question hide_question_numbers=1]174228[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Solve inverse variation problems<\/h2>\r\n<p id=\"fs-id1165137734583\">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>\r\n<p id=\"fs-id1165137761800\">If we create a table\u00a0we observe that, as the depth increases, the water temperature decreases.<\/p>\r\n\r\n<table id=\"Table_03_09_02\" 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\n<p id=\"fs-id1165137645896\">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>\r\n<p id=\"fs-id1165137805474\">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>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010806\/CNX_Precalc_Figure_03_09_0032.jpg\" alt=\"Graph of y=(14000)\/x where the horizontal axis is labeled, \" width=\"487\" height=\"309\" \/> <b>Figure 3<\/b>[\/caption]\r\n\r\n<div id=\"fs-id1165135397976\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Inverse Variation<\/h3>\r\n<p id=\"fs-id1165137536242\">If <em>x<\/em>\u00a0and <em>y<\/em>\u00a0are related by an equation of the form<\/p>\r\n\r\n<div id=\"fs-id1165137571596\" class=\"equation\" style=\"text-align: center;\">[latex]y=\\frac{k}{{x}^{n}}[\/latex]<\/div>\r\n<p id=\"fs-id1165137843973\">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>\r\n\r\n<\/div>\r\n<div id=\"Example_03_09_02\" class=\"example\">\r\n<div id=\"fs-id1165137641735\" class=\"exercise\">\r\n<div id=\"fs-id1165137658061\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Writing a Formula for an Inversely Proportional Relationship<\/h3>\r\n<p id=\"fs-id1165131797298\">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>\r\n[reveal-answer q=\"363929\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"363929\"]\r\n<p id=\"fs-id1165137827766\">Recall 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>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}t\\left(v\\right)&amp;=\\frac{100}{v} \\\\[1mm] &amp;=100{v}^{-1} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137748472\">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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135187117\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137677962\">How To: Given a description of an indirect variation problem, solve for an unknown.<strong>\r\n<\/strong><\/h3>\r\n<ol id=\"fs-id1165137662822\">\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 id=\"Example_03_09_03\" class=\"example\">\r\n<div id=\"fs-id1165134328944\" class=\"exercise\">\r\n<div id=\"fs-id1165137581324\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Solving an Inverse Variation Problem<\/h3>\r\n<p id=\"fs-id1165135209804\">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>\r\n[reveal-answer q=\"251565\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"251565\"]\r\n<p id=\"fs-id1165137627457\">The 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>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}k&amp;={x}^{3}y \\\\ &amp;={2}^{3}\\cdot 25 \\\\ &amp;=200 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135188786\">Now we use the constant to write an equation that represents this relationship.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;y=\\frac{k}{{x}^{3}},k=200 \\\\ &amp;y=\\frac{200}{{x}^{3}} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137653904\">Substitute <em>x\u00a0<\/em>= 6 and solve for <i>y<\/i>.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\frac{200}{{6}^{3}} \\\\ &amp;=\\frac{25}{27} \\end{align}[\/latex]<\/p>\r\n\r\n<h3>Analysis of the Solution<\/h3>\r\n<p id=\"fs-id1165137852181\">The graph of this equation is a rational function.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010806\/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\" \/> <b>Figure 4<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137573081\" class=\"commentary\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137810878\">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>\r\n[reveal-answer q=\"595348\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"595348\"]\r\n\r\n[latex]\\frac{9}{2}[\/latex]\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\n[ohm_question hide_question_numbers=1]14328[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Solve problems involving joint variation<\/h2>\r\n<p id=\"fs-id1165137558033\">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>\r\n\r\n<div id=\"fs-id1165135177639\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Joint Variation<\/h3>\r\n<p id=\"fs-id1165135195246\">Joint variation occurs when a variable varies directly or inversely with multiple variables.<\/p>\r\n<p id=\"fs-id1165137678943\">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>\r\n\r\n<\/div>\r\n<div id=\"Example_03_09_04\" class=\"example\">\r\n<div id=\"fs-id1165137673524\" class=\"exercise\">\r\n<div id=\"fs-id1165135394333\" class=\"problem textbox shaded\">\r\n<h3>Example 4: Solving Problems Involving Joint Variation<\/h3>\r\n<p id=\"fs-id1165137452990\">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>\r\n[reveal-answer q=\"529237\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"529237\"]\r\n<p id=\"fs-id1165133213902\">Begin by writing an equation to show the relationship between the variables.<\/p>\r\n<p style=\"text-align: center;\">[latex]x=\\frac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\r\n<p id=\"fs-id1165135190190\">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>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;6=\\frac{k{2}^{2}}{\\sqrt[3]{8}} \\\\ &amp;6=\\frac{4k}{2} \\\\ &amp;3=k \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137863719\">Now we can substitute the value of the constant into the equation for the relationship.<\/p>\r\n<p style=\"text-align: center;\">[latex]x=\\frac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\r\n<p id=\"fs-id1165137742401\">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>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x&amp;=\\frac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}} \\\\ &amp;=1 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165137588086\"><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>\r\n[reveal-answer q=\"359999\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"359999\"]\r\n\r\n<em>x<\/em> = 20\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Takeaways<\/h3>\r\n[ohm_question sameseed=1 hide_question_numbers=1]40895[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #1d1d1d; font-size: 1.5em; font-weight: bold;\">Key Equations<\/span>\r\n\r\n<section id=\"fs-id1165137898092\" class=\"key-equations\">\r\n<table id=\"eip-id1165133094986\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>Direct variation<\/td>\r\n<td>[latex]y=k{x}^{n},k\\text{ is a nonzero constant}[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Inverse variation<\/td>\r\n<td>[latex]y=\\frac{k}{{x}^{n}},k\\text{ is a nonzero constant}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165137419773\" class=\"key-concepts\">\r\n<h1>Key Concepts<\/h1>\r\n<ul id=\"fs-id1165137723142\">\r\n \t<li>A relationship where one quantity is a constant multiplied by another quantity is called direct variation.<\/li>\r\n \t<li>Two variables that are directly proportional to one another will have a constant ratio.<\/li>\r\n \t<li>A relationship where one quantity is a constant divided by another quantity is called inverse variation.<\/li>\r\n \t<li>Two variables that are inversely proportional to one another will have a constant multiple.<\/li>\r\n \t<li>In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137735724\" class=\"definition\">\r\n \t<dt><strong>constant of variation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137735729\">the non-zero value <em>k<\/em>\u00a0that helps define the relationship between variables in direct or inverse variation<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137762202\" class=\"definition\">\r\n \t<dt><strong>direct variation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137762208\">the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137462046\" class=\"definition\">\r\n \t<dt><strong>inverse variation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137462052\">the relationship between two variables in which the product of the variables is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135501040\" class=\"definition\">\r\n \t<dt><strong>inversely proportional<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137874542\">a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137874546\" class=\"definition\">\r\n \t<dt><strong>joint variation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135696715\">a relationship where a variable varies directly or inversely with multiple variables<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135696718\" class=\"definition\">\r\n \t<dt><strong>varies directly<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137432955\">a relationship where one quantity is a constant multiplied by the other quantity<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137432958\" class=\"definition\">\r\n \t<dt><strong>varies inversely<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135439853\">a relationship where one quantity is a constant divided by the other quantity<\/dd>\r\n<\/dl>\r\n<\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\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<\/div>\n<p id=\"fs-id1165135356540\">A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.<\/p>\n<h2>Solve direct variation problems<\/h2>\n<p id=\"fs-id1165137823230\">In the example above, Nicole\u2019s earnings can be found by multiplying her sales by her commission. The formula <em>e<\/em> = 0.16<em>s<\/em> tells us her earnings, <em>e<\/em>, come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.<\/p>\n<table id=\"Table_03_09_01\" summary=\"..\">\n<thead>\n<tr>\n<th><em>s<\/em>, sales prices<\/th>\n<th><em>e<\/em> = 0.16<em>s<\/em><\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>$4,600<\/td>\n<td><em>e\u00a0<\/em>= 0.16(4,600) = 736<\/td>\n<td>A sale of a $4,600 vehicle results in $736 earnings.<\/td>\n<\/tr>\n<tr>\n<td>$9,200<\/td>\n<td><em>e\u00a0<\/em>= 0.16(9,200) = 1,472<\/td>\n<td>A sale of a $9,200 vehicle results in $1472 earnings.<\/td>\n<\/tr>\n<tr>\n<td>$18,400<\/td>\n<td><em>e\u00a0<\/em>= 0.16(18,400) = 2,944<\/td>\n<td>A sale of a $18,400 vehicle results in $2944 earnings.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135188294\">Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called <strong>direct variation<\/strong>. Each variable in this type of relationship <strong>varies directly <\/strong>with the other.<\/p>\n<p id=\"fs-id1165137937533\">The graph below\u00a0represents the data for Nicole\u2019s potential earnings. We say that earnings vary directly with the sales price of the car. The formula [latex]y=k{x}^{n}[\/latex] is used for direct variation. The value <em>k<\/em>\u00a0is a nonzero constant greater than zero and is called the <strong>constant of variation<\/strong>. In this case, <em>k\u00a0<\/em>= 0.16 and <em>n\u00a0<\/em>= 1.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010805\/CNX_Precalc_Figure_03_09_0012.jpg\" alt=\"Graph of y=(0.16)x where the horizontal axis is labeled,\" width=\"487\" height=\"459\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<div id=\"fs-id1165137730075\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Direct Variation<\/h3>\n<p id=\"fs-id1165137827458\">If <em>x <\/em>and <em>y<\/em>\u00a0are related by an equation of the form<\/p>\n<div id=\"fs-id1165135437156\" class=\"equation\" style=\"text-align: center;\">[latex]y=k{x}^{n}[\/latex]<\/div>\n<p id=\"fs-id1165133155266\">then we say that the relationship is <strong>direct variation<\/strong> and <em>y<\/em>\u00a0<strong>varies directly<\/strong> with the <em>n<\/em>th power of <em>x<\/em>. In direct variation relationships, there is a nonzero constant ratio [latex]k=\\frac{y}{{x}^{n}}[\/latex], where <em>k<\/em>\u00a0is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.<\/p>\n<\/div>\n<div id=\"fs-id1165137550958\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137723932\">How To: Given a description of a direct variation problem, solve for an unknown.<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1165137724401\">\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 divide <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 id=\"Example_03_09_01\" class=\"example\">\n<div id=\"fs-id1165137676066\" class=\"exercise\">\n<div id=\"fs-id1165137434564\" class=\"problem textbox shaded\">\n<h3>Example 1: Solving a Direct Variation Problem<\/h3>\n<p id=\"fs-id1165137849016\">The quantity <em>y<\/em>\u00a0varies directly 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=\"q467589\">Show Solution<\/span><\/p>\n<div id=\"q467589\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137659713\">The general formula for direct variation with a cube is [latex]y=k{x}^{3}[\/latex]. The constant can be found by dividing <em>y<\/em>\u00a0by the cube of <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} k&=\\frac{y}{{x}^{3}} \\\\[1mm] &=\\frac{25}{{2}^{3}}\\\\[1mm] &=\\frac{25}{8}\\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137628102\">Now use the constant to write an equation that represents this relationship.<\/p>\n<p style=\"text-align: center;\">[latex]y=\\frac{25}{8}{x}^{3}[\/latex]<\/p>\n<p id=\"fs-id1165135432964\">Substitute <em>x<\/em> = 6 and solve for <em>y<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&=\\frac{25}{8}{\\left(6\\right)}^{3} \\\\[1mm] &=675 \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165134557390\">The graph of this equation is a simple cubic, as shown below.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010805\/CNX_Precalc_Figure_03_09_0022.jpg\" alt=\"Graph of y=25\/8(x^3) with the labeled points (2, 25) and (6, 675).\" width=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135533140\" class=\"commentary\">\n<div class=\"mceTemp\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736204\" class=\"note precalculus qa textbox\">\n<h3>Q &amp; A<\/h3>\n<p id=\"eip-id1165137772190\"><strong>Do the graphs of all direct variation equations look like Example 1?<\/strong><\/p>\n<p id=\"fs-id1165137596402\"><em>No. Direct variation equations are power functions\u2014they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through (0, 0).<\/em><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135160334\">The quantity <em>y<\/em>\u00a0varies directly with the square of <em>x<\/em>. If <em>y\u00a0<\/em>= 24 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=\"q272502\">Show Solution<\/span><\/p>\n<div id=\"q272502\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{128}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174228\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174228&theme=oea&iframe_resize_id=ohm174228\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Solve inverse variation problems<\/h2>\n<p id=\"fs-id1165137734583\">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 id=\"fs-id1165137761800\">If we create a table\u00a0we observe that, as the depth increases, the water temperature decreases.<\/p>\n<table id=\"Table_03_09_02\" 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 id=\"fs-id1165137645896\">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 id=\"fs-id1165137805474\">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<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010806\/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<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<div id=\"fs-id1165135397976\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Inverse Variation<\/h3>\n<p id=\"fs-id1165137536242\">If <em>x<\/em>\u00a0and <em>y<\/em>\u00a0are related by an equation of the form<\/p>\n<div id=\"fs-id1165137571596\" class=\"equation\" style=\"text-align: center;\">[latex]y=\\frac{k}{{x}^{n}}[\/latex]<\/div>\n<p id=\"fs-id1165137843973\">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 id=\"Example_03_09_02\" class=\"example\">\n<div id=\"fs-id1165137641735\" class=\"exercise\">\n<div id=\"fs-id1165137658061\" class=\"problem textbox shaded\">\n<h3>Example 2: Writing a Formula for an Inversely Proportional Relationship<\/h3>\n<p id=\"fs-id1165131797298\">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=\"q363929\">Show Solution<\/span><\/p>\n<div id=\"q363929\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137827766\">Recall 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{align}t\\left(v\\right)&=\\frac{100}{v} \\\\[1mm] &=100{v}^{-1} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137748472\">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>\n<\/div>\n<div id=\"fs-id1165135187117\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137677962\">How To: Given a description of an indirect variation problem, solve for an unknown.<strong><br \/>\n<\/strong><\/h3>\n<ol id=\"fs-id1165137662822\">\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 id=\"Example_03_09_03\" class=\"example\">\n<div id=\"fs-id1165134328944\" class=\"exercise\">\n<div id=\"fs-id1165137581324\" class=\"problem textbox shaded\">\n<h3>Example 3: Solving an Inverse Variation Problem<\/h3>\n<p id=\"fs-id1165135209804\">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=\"q251565\">Show Solution<\/span><\/p>\n<div id=\"q251565\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137627457\">The 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{align}k&={x}^{3}y \\\\ &={2}^{3}\\cdot 25 \\\\ &=200 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135188786\">Now we use the constant to write an equation that represents this relationship.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&y=\\frac{k}{{x}^{3}},k=200 \\\\ &y=\\frac{200}{{x}^{3}} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137653904\">Substitute <em>x\u00a0<\/em>= 6 and solve for <i>y<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&=\\frac{200}{{6}^{3}} \\\\ &=\\frac{25}{27} \\end{align}[\/latex]<\/p>\n<h3>Analysis of the Solution<\/h3>\n<p id=\"fs-id1165137852181\">The graph of this equation is a rational function.<\/p>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010806\/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<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137573081\" class=\"commentary\"><\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137810878\">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=\"q595348\">Show Solution<\/span><\/p>\n<div id=\"q595348\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{9}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm14328\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14328&theme=oea&iframe_resize_id=ohm14328\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Solve problems involving joint variation<\/h2>\n<p id=\"fs-id1165137558033\">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 id=\"fs-id1165135177639\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Joint Variation<\/h3>\n<p id=\"fs-id1165135195246\">Joint variation occurs when a variable varies directly or inversely with multiple variables.<\/p>\n<p id=\"fs-id1165137678943\">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 id=\"Example_03_09_04\" class=\"example\">\n<div id=\"fs-id1165137673524\" class=\"exercise\">\n<div id=\"fs-id1165135394333\" class=\"problem textbox shaded\">\n<h3>Example 4: Solving Problems Involving Joint Variation<\/h3>\n<p id=\"fs-id1165137452990\">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=\"q529237\">Show Solution<\/span><\/p>\n<div id=\"q529237\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165133213902\">Begin 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 id=\"fs-id1165135190190\">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{align}&6=\\frac{k{2}^{2}}{\\sqrt[3]{8}} \\\\ &6=\\frac{4k}{2} \\\\ &3=k \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137863719\">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 id=\"fs-id1165137742401\">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{align}x&=\\frac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}} \\\\ &=1 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137588086\"><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=\"q359999\">Show Solution<\/span><\/p>\n<div id=\"q359999\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>x<\/em> = 20<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Takeaways<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm40895\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40895&theme=oea&iframe_resize_id=ohm40895&sameseed=1\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #1d1d1d; font-size: 1.5em; font-weight: bold;\">Key Equations<\/span><\/p>\n<section id=\"fs-id1165137898092\" class=\"key-equations\">\n<table id=\"eip-id1165133094986\" summary=\"..\">\n<tbody>\n<tr>\n<td>Direct variation<\/td>\n<td>[latex]y=k{x}^{n},k\\text{ is a nonzero constant}[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Inverse variation<\/td>\n<td>[latex]y=\\frac{k}{{x}^{n}},k\\text{ is a nonzero constant}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137419773\" class=\"key-concepts\">\n<h1>Key Concepts<\/h1>\n<ul id=\"fs-id1165137723142\">\n<li>A relationship where one quantity is a constant multiplied by another quantity is called direct variation.<\/li>\n<li>Two variables that are directly proportional to one another will have a constant ratio.<\/li>\n<li>A relationship where one quantity is a constant divided by another quantity is called inverse variation.<\/li>\n<li>Two variables that are inversely proportional to one another will have a constant multiple.<\/li>\n<li>In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137735724\" class=\"definition\">\n<dt><strong>constant of variation<\/strong><\/dt>\n<dd id=\"fs-id1165137735729\">the non-zero value <em>k<\/em>\u00a0that helps define the relationship between variables in direct or inverse variation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137762202\" class=\"definition\">\n<dt><strong>direct variation<\/strong><\/dt>\n<dd id=\"fs-id1165137762208\">the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137462046\" class=\"definition\">\n<dt><strong>inverse variation<\/strong><\/dt>\n<dd id=\"fs-id1165137462052\">the relationship between two variables in which the product of the variables is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135501040\" class=\"definition\">\n<dt><strong>inversely proportional<\/strong><\/dt>\n<dd id=\"fs-id1165137874542\">a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137874546\" class=\"definition\">\n<dt><strong>joint variation<\/strong><\/dt>\n<dd id=\"fs-id1165135696715\">a relationship where a variable varies directly or inversely with multiple variables<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135696718\" class=\"definition\">\n<dt><strong>varies directly<\/strong><\/dt>\n<dd id=\"fs-id1165137432955\">a relationship where one quantity is a constant multiplied by the other quantity<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137432958\" class=\"definition\">\n<dt><strong>varies inversely<\/strong><\/dt>\n<dd id=\"fs-id1165135439853\">a relationship where one quantity is a constant divided by the other quantity<\/dd>\n<\/dl>\n<\/section>\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-13901\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/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":97803,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-13901","chapter","type-chapter","status-publish","hentry"],"part":10733,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13901","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/97803"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13901\/revisions"}],"predecessor-version":[{"id":15844,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13901\/revisions\/15844"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/10733"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/13901\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=13901"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=13901"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=13901"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=13901"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}