{"id":5309,"date":"2021-10-11T20:36:56","date_gmt":"2021-10-11T20:36:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/chapter\/direct-variation\/"},"modified":"2022-03-30T02:01:24","modified_gmt":"2022-03-30T02:01:24","slug":"direct-variation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lcudd-tulsacc-collegealgebra\/chapter\/direct-variation\/","title":{"raw":"Variation","rendered":"Variation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve a direct variation problem<\/li>\r\n \t<li>Use a constant of variation to describe the relationship between two variables<\/li>\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 \t<li>Solve a Joint variation problem.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Direct Variation<\/h2>\r\nA 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.\r\n\r\nIn the example above, Nicole\u2019s earnings can be found by multiplying her sales by her commission. The formula [latex]e = 0.16s[\/latex] tells us her earnings, [latex]e[\/latex], come from the product of 0.16, her commission, and the sale price of the vehicle, [latex]s[\/latex]. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.\r\n<table summary=\"..\">\r\n<thead>\r\n<tr>\r\n<th>[latex]s[\/latex], sales prices<\/th>\r\n<th>[latex]e = 0.16s[\/latex]<\/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>[latex]e=0.16(4,600)=736[\/latex]<\/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>[latex]e=0.16(9,200)=1,472[\/latex]<\/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>[latex]e=0.16(18,400)=2,944[\/latex]<\/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\nNotice 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.\u00a0 You could also say that one quantity is <strong>directly proportional<\/strong> to another quantity.\r\n\r\nThe 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 [latex]k[\/latex] is a nonzero constant greater than zero and is called the <strong>constant of variation<\/strong>. In this case, [latex]k=0.16[\/latex]\u00a0and [latex]n=1[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222950\/CNX_Precalc_Figure_03_09_0012.jpg\" alt=\"Graph of y=(0.16)x where the horizontal axis is labeled,\" width=\"487\" height=\"459\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Direct Variation<\/h3>\r\nIf [latex]x[\/latex]<em>\u00a0<\/em>and [latex]y[\/latex]\u00a0are related by an equation of the form\r\n<p style=\"text-align: center;\">[latex]y=k{x}^{n}[\/latex]<\/p>\r\nthen we say that the relationship is <strong>direct variation<\/strong> and [latex]y[\/latex]\u00a0<strong>varies directly<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex]. In direct variation relationships, there is a nonzero constant ratio [latex]k=\\dfrac{y}{{x}^{n}}[\/latex], where [latex]k[\/latex]\u00a0is called the <strong>constant of variation<\/strong>, which helps define the relationship between the variables.\r\n\r\nDoes the direct variation equation above look familiar?\u00a0 Direct variation equations are <strong>power functions<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>recall isolating a variable in a formula<\/h3>\r\nWe've learned to solve certain formulas for one of the variables. For example, in the formula that relates distance, rate, and time, [latex]d=rt[\/latex], we can solve the equation for rate, [latex]r=\\dfrac{d}{t}[\/latex] or for time, [latex]t=\\dfrac{d}{r}[\/latex]. We say we are\u00a0<em>isolating\u00a0<\/em>the variable of interest in these cases.\r\n\r\nThe same idea applies when solving a direct variation problem for the constant of variation, [latex]k[\/latex]. Given a direct variation such as [latex]y=kx^n[\/latex], we can <em>isolate<\/em> the constant of variation using the properties of equality to obtain [latex]\\dfrac{y}{x^n}=k[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>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, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\r\n \t<li>Determine the constant of variation. You may need to divide [latex]y[\/latex]\u00a0by the specified power of [latex]x[\/latex]\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 a Direct Variation Problem<\/h3>\r\nThe quantity [latex]y[\/latex]\u00a0varies directly with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]\u00a0when [latex]x=2[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 6.\r\n\r\n[reveal-answer q=\"647220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"647220\"]\r\n\r\nThe general formula for direct variation with a cube is [latex]y=k{x}^{3}[\/latex]. The constant can be found by dividing [latex]y[\/latex]\u00a0by the cube of [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align} k&amp;=\\dfrac{y}{{x}^{3}} \\\\[1mm] &amp;=\\dfrac{25}{{2}^{3}}\\\\[1mm] &amp;=\\dfrac{25}{8}\\end{align}[\/latex]<\/p>\r\nNow use the constant to write an equation that represents this relationship.\r\n<p style=\"text-align: center;\">[latex]y=\\dfrac{25}{8}{x}^{3}[\/latex]<\/p>\r\nSubstitute [latex]x=6[\/latex]\u00a0and solve for [latex]y[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\dfrac{25}{8}{\\left(6\\right)}^{3} \\\\[1mm] &amp;=675\\hfill \\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThe graph of this equation is a simple cubic, as shown below.\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222952\/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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Do the graphs of all direct variation equations look like the example above?<\/strong>\r\n\r\n<em>No. Direct variation equations are power functions\u2014they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through <\/em>[latex](0, 0)[\/latex]<em>.<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe quantity [latex]y[\/latex]\u00a0varies directly with the square of [latex]x[\/latex]. If [latex]y=24[\/latex]\u00a0when [latex]x=3[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 4.\r\n\r\n[reveal-answer q=\"536994\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"536994\"]\r\n\r\n[latex]\\dfrac{128}{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<strong>In the problem below, recall that \"directly proportional\" is another way to refer to direct variation.<\/strong>\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91391&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\r\n\r\n<\/div>\r\nWatch this video to see a quick lesson in direct variation. \u00a0You will see more worked examples.\u00a0 Again, the phrase \"directly proportional\" is another way to refer to \"direct variation\".\r\n\r\nhttps:\/\/youtu.be\/plFOq4JaEyI\r\n<h2>Inverse Variation<\/h2>\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>[latex]d[\/latex], 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=\\dfrac{k}{x}[\/latex] for inverse variation in this case uses [latex]k=14,000[\/latex].\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\" \/>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Inverse Variation<\/h3>\r\nIf [latex]x[\/latex] and [latex]y[\/latex]\u00a0are related by an equation of the form\r\n\r\n[latex]y=\\dfrac{k}{{x}^{n}}[\/latex]\r\n\r\nwhere [latex]k[\/latex]\u00a0is a nonzero constant, then we say that [latex]y[\/latex]\u00a0<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].\u00a0 Even though we normally write inverse variation equations like\u00a0[latex]y=\\dfrac{k}{{x}^{n}}[\/latex], these equations are also power functions if we rewrite them using negative exponents\u00a0[latex]y=k{x}^{-n}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>isolating the constant of variation<\/h3>\r\nTo isolate the constant of variation in an inverse variation, use the properties of equality to solve the equation for [latex]k[\/latex].\r\n\r\n[latex]y=\\dfrac{k}{{x}^{n}}[\/latex]\r\n\r\nIsolate [latex]k[\/latex] using algebra.\r\n\r\n[latex]yx^n=k[\/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\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"81111\"]\r\n\r\nRecall that multiplying speed by time gives distance. If we let [latex]t[\/latex]\u00a0represent the drive time in hours, and [latex]v[\/latex]\u00a0represent the velocity (speed or rate) at which the tourist drives, then [latex]vt=[\/latex]\u00a0distance. Because the distance is fixed at 100 miles, [latex]vt=100[\/latex]. Solving this relationship for the time gives us our function.\r\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>\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 inverse variation problem, solve for an unknown.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\r\n \t<li>Determine the constant of variation. You may need to multiply [latex]y[\/latex]\u00a0by the specified power of [latex]x[\/latex]\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 [latex]y[\/latex]\u00a0varies inversely with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]\u00a0when [latex]x=2[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 6.\r\n\r\n[reveal-answer q=\"482072\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"482072\"]\r\n\r\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]\u00a0by the cube of [latex]x[\/latex].\r\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>\r\nNow we use the constant to write an equation that represents this relationship.\r\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>\r\nSubstitute [latex]x=6[\/latex]\u00a0and solve for [latex]y[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\dfrac{200}{{6}^{3}} \\\\[1mm] &amp;=\\dfrac{25}{27} \\end{align}[\/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\" \/>\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 [latex]y[\/latex]\u00a0varies inversely with the square of [latex]x[\/latex]. If [latex]y=8[\/latex]\u00a0when [latex]x=3[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 4.\r\n\r\n[reveal-answer q=\"285259\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"285259\"]\r\n\r\n[latex]\\dfrac{9}{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91393&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\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 [latex]c[\/latex], cost, varies jointly with the number of students, [latex]n[\/latex], and the distance, [latex]d[\/latex].\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 [latex]x[\/latex]\u00a0varies directly with both [latex]y[\/latex]\u00a0and [latex]z[\/latex], we have [latex]x=kyz[\/latex]. If [latex]x[\/latex]\u00a0varies directly with [latex]y[\/latex]\u00a0and 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.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>isolating the constant of variation<\/h3>\r\nTo isolate the constant of variation in a joint variation, use the properties of equality to solve the equation for [latex]k[\/latex].\r\n\r\n[latex]x=kyz[\/latex]\r\n\r\nIsolate [latex]k[\/latex] using algebra.\r\n\r\n[latex]\\dfrac{x}{yz}=k[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving Problems Involving Joint Variation<\/h3>\r\nA quantity [latex]x[\/latex]\u00a0varies directly with the square of [latex]y[\/latex]\u00a0and inversely with the cube root of [latex]z[\/latex]. If [latex]x=6[\/latex]\u00a0when [latex]y=2[\/latex]\u00a0and [latex]z=8[\/latex], find [latex]x[\/latex]\u00a0when [latex]y=1[\/latex]\u00a0and [latex]z=27[\/latex].\r\n\r\n[reveal-answer q=\"396823\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"396823\"]\r\n\r\nBegin by writing an equation to show the relationship between the variables.\r\n<p style=\"text-align: center;\">[latex]x=\\dfrac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\r\nSubstitute [latex]x=6[\/latex], [latex]y=2[\/latex], and [latex]z=8[\/latex]\u00a0to find the value of the constant [latex]k[\/latex].\r\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>\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=\\dfrac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\r\nTo find [latex]x[\/latex]\u00a0when [latex]y=1[\/latex]\u00a0and [latex]z=27[\/latex], we will substitute values for [latex]y[\/latex]\u00a0and [latex]z[\/latex]\u00a0into our equation.\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[latex]x[\/latex] varies directly with the square of [latex]y[\/latex]\u00a0and inversely with [latex]z[\/latex]. If [latex]x=40[\/latex]\u00a0when [latex]y=4[\/latex]\u00a0and [latex]z=2[\/latex], find [latex]x[\/latex]\u00a0when [latex]y=10[\/latex]\u00a0and [latex]z=25[\/latex].\r\n\r\n[reveal-answer q=\"286100\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"286100\"]\r\n\r\n[latex]x=20[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91394&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\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 Outcomes<\/h3>\n<ul>\n<li>Solve a direct variation problem<\/li>\n<li>Use a constant of variation to describe the relationship between two variables<\/li>\n<li>Solve an Inverse variation problem.<\/li>\n<li>Write a formula for an inversely proportional relationship.<\/li>\n<li>Solve a Joint variation problem.<\/li>\n<\/ul>\n<\/div>\n<h2>Direct Variation<\/h2>\n<p>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<p>In the example above, Nicole\u2019s earnings can be found by multiplying her sales by her commission. The formula [latex]e = 0.16s[\/latex] tells us her earnings, [latex]e[\/latex], come from the product of 0.16, her commission, and the sale price of the vehicle, [latex]s[\/latex]. 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 summary=\"..\">\n<thead>\n<tr>\n<th>[latex]s[\/latex], sales prices<\/th>\n<th>[latex]e = 0.16s[\/latex]<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>$4,600<\/td>\n<td>[latex]e=0.16(4,600)=736[\/latex]<\/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>[latex]e=0.16(9,200)=1,472[\/latex]<\/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>[latex]e=0.16(18,400)=2,944[\/latex]<\/td>\n<td>A sale of a $18,400 vehicle results in $2944 earnings.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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.\u00a0 You could also say that one quantity is <strong>directly proportional<\/strong> to another quantity.<\/p>\n<p>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 [latex]k[\/latex] is a nonzero constant greater than zero and is called the <strong>constant of variation<\/strong>. In this case, [latex]k=0.16[\/latex]\u00a0and [latex]n=1[\/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\/02222950\/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<div class=\"textbox\">\n<h3>A General Note: Direct Variation<\/h3>\n<p>If [latex]x[\/latex]<em>\u00a0<\/em>and [latex]y[\/latex]\u00a0are related by an equation of the form<\/p>\n<p style=\"text-align: center;\">[latex]y=k{x}^{n}[\/latex]<\/p>\n<p>then we say that the relationship is <strong>direct variation<\/strong> and [latex]y[\/latex]\u00a0<strong>varies directly<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex]. In direct variation relationships, there is a nonzero constant ratio [latex]k=\\dfrac{y}{{x}^{n}}[\/latex], where [latex]k[\/latex]\u00a0is called the <strong>constant of variation<\/strong>, which helps define the relationship between the variables.<\/p>\n<p>Does the direct variation equation above look familiar?\u00a0 Direct variation equations are <strong>power functions<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>recall isolating a variable in a formula<\/h3>\n<p>We&#8217;ve learned to solve certain formulas for one of the variables. For example, in the formula that relates distance, rate, and time, [latex]d=rt[\/latex], we can solve the equation for rate, [latex]r=\\dfrac{d}{t}[\/latex] or for time, [latex]t=\\dfrac{d}{r}[\/latex]. We say we are\u00a0<em>isolating\u00a0<\/em>the variable of interest in these cases.<\/p>\n<p>The same idea applies when solving a direct variation problem for the constant of variation, [latex]k[\/latex]. Given a direct variation such as [latex]y=kx^n[\/latex], we can <em>isolate<\/em> the constant of variation using the properties of equality to obtain [latex]\\dfrac{y}{x^n}=k[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>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, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\n<li>Determine the constant of variation. You may need to divide [latex]y[\/latex]\u00a0by the specified power of [latex]x[\/latex]\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 a Direct Variation Problem<\/h3>\n<p>The quantity [latex]y[\/latex]\u00a0varies directly with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]\u00a0when [latex]x=2[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 6.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q647220\">Show Solution<\/span><\/p>\n<div id=\"q647220\" class=\"hidden-answer\" style=\"display: none\">\n<p>The general formula for direct variation with a cube is [latex]y=k{x}^{3}[\/latex]. The constant can be found by dividing [latex]y[\/latex]\u00a0by the cube of [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} k&=\\dfrac{y}{{x}^{3}} \\\\[1mm] &=\\dfrac{25}{{2}^{3}}\\\\[1mm] &=\\dfrac{25}{8}\\end{align}[\/latex]<\/p>\n<p>Now use the constant to write an equation that represents this relationship.<\/p>\n<p style=\"text-align: center;\">[latex]y=\\dfrac{25}{8}{x}^{3}[\/latex]<\/p>\n<p>Substitute [latex]x=6[\/latex]\u00a0and solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&=\\dfrac{25}{8}{\\left(6\\right)}^{3} \\\\[1mm] &=675\\hfill \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph of this equation is a simple cubic, as shown below.<\/p>\n<p>&nbsp;<\/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\/02222952\/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<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Do the graphs of all direct variation equations look like the example above?<\/strong><\/p>\n<p><em>No. Direct variation equations are power functions\u2014they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through <\/em>[latex](0, 0)[\/latex]<em>.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The quantity [latex]y[\/latex]\u00a0varies directly with the square of [latex]x[\/latex]. If [latex]y=24[\/latex]\u00a0when [latex]x=3[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 4.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q536994\">Show Solution<\/span><\/p>\n<div id=\"q536994\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{128}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><strong>In the problem below, recall that &#8220;directly proportional&#8221; is another way to refer to direct variation.<\/strong><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm91391\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91391&#38;theme=oea&#38;iframe_resize_id=ohm91391&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see a quick lesson in direct variation. \u00a0You will see more worked examples.\u00a0 Again, the phrase &#8220;directly proportional&#8221; is another way to refer to &#8220;direct variation&#8221;.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Direct Variation Applications\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/plFOq4JaEyI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Inverse Variation<\/h2>\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>[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\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=\\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]\u00a0are related by an equation of the form<\/p>\n<p>[latex]y=\\dfrac{k}{{x}^{n}}[\/latex]<\/p>\n<p>where [latex]k[\/latex]\u00a0is a nonzero constant, then we say that [latex]y[\/latex]\u00a0<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].\u00a0 Even though we normally write inverse variation equations like\u00a0[latex]y=\\dfrac{k}{{x}^{n}}[\/latex], these equations are also power functions if we rewrite them using negative exponents\u00a0[latex]y=k{x}^{-n}[\/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]\u00a0represent the drive time in hours, and [latex]v[\/latex]\u00a0represent the velocity (speed or rate) at which the tourist drives, then [latex]vt=[\/latex]\u00a0distance. 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]\u00a0by the specified power of [latex]x[\/latex]\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 [latex]y[\/latex]\u00a0varies inversely with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]\u00a0when [latex]x=2[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 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]\u00a0by 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]\u00a0and 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]\u00a0varies inversely with the square of [latex]x[\/latex]. If [latex]y=8[\/latex]\u00a0when [latex]x=3[\/latex], find [latex]y[\/latex]\u00a0when [latex]x[\/latex]\u00a0is 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-2\" 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]\u00a0varies directly with both [latex]y[\/latex]\u00a0and [latex]z[\/latex], we have [latex]x=kyz[\/latex]. If [latex]x[\/latex]\u00a0varies directly with [latex]y[\/latex]\u00a0and 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]\u00a0varies directly with the square of [latex]y[\/latex]\u00a0and inversely with the cube root of [latex]z[\/latex]. If [latex]x=6[\/latex]\u00a0when [latex]y=2[\/latex]\u00a0and [latex]z=8[\/latex], find [latex]x[\/latex]\u00a0when [latex]y=1[\/latex]\u00a0and [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]\u00a0to 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]\u00a0when [latex]y=1[\/latex]\u00a0and [latex]z=27[\/latex], we will substitute values for [latex]y[\/latex]\u00a0and [latex]z[\/latex]\u00a0into 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]\u00a0and inversely with [latex]z[\/latex]. If [latex]x=40[\/latex]\u00a0when [latex]y=4[\/latex]\u00a0and [latex]z=2[\/latex], find [latex]x[\/latex]\u00a0when [latex]y=10[\/latex]\u00a0and [latex]z=25[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><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-3\" 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-5309\">\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 91391. <strong>Authored by<\/strong>: Jenck, Michael. <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>Direct Variation Applications . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/plFOq4JaEyI\">https:\/\/youtu.be\/plFOq4JaEyI<\/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":167848,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 91391\",\"author\":\"Jenck, Michael\",\"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 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