{"id":1876,"date":"2023-10-12T00:32:23","date_gmt":"2023-10-12T00:32:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-variation\/"},"modified":"2023-10-12T00:32:23","modified_gmt":"2023-10-12T00:32:23","slug":"introduction-variation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-variation\/","title":{"raw":"Variation","rendered":"Variation"},"content":{"raw":"\n\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n \t<li class=\"li2\"><span class=\"s1\">Solve direct variation problems.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Solve inverse variation problems.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Solve problems involving joint variation.<\/span><\/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\n<h2>Direct Variation<\/h2>\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].&nbsp;If we create a table, we observe that as the sales price increases, the earnings increase as well.\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<em>&nbsp;<\/em>= 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<em>&nbsp;<\/em>= 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<em>&nbsp;<\/em>= 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>\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.\n\nThe graph below&nbsp;represents 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]&nbsp;and [latex]n=1[\/latex].\n\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\">\n<div class=\"textbox\">\n<h3>A General Note: Direct Variation<\/h3>\nIf [latex]x[\/latex]<em>&nbsp;<\/em>and [latex]y[\/latex]&nbsp;are related by an equation of the form\n<p style=\"text-align: center;\">[latex]y=k{x}^{n}[\/latex]<\/p>\nthen we say that the relationship is <strong>direct variation<\/strong> and [latex]y[\/latex]&nbsp;<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]&nbsp;is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a description of a direct variation problem, solve for an unknown.<strong>\n<\/strong><\/h3>\n<ol id=\"fs-id1165137724401\">\n \t<li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\n \t<li>Determine the constant of variation. You may need to divide [latex]y[\/latex]&nbsp;by the specified power of [latex]x[\/latex]&nbsp;to determine the constant of variation.<\/li>\n \t<li>Use the constant of variation to write an equation for the relationship.<\/li>\n \t<li>Substitute known values into the equation to find the unknown.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Direct Variation Problem<\/h3>\nThe quantity [latex]y[\/latex]&nbsp;varies directly with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]&nbsp;when [latex]x=2[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 6.\n\n[reveal-answer q=\"647220\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"647220\"]\n\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]&nbsp;by the cube of [latex]x[\/latex].\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>\nNow use the constant to write an equation that represents this relationship.\n<p style=\"text-align: center;\">[latex]y=\\dfrac{25}{8}{x}^{3}[\/latex]<\/p>\nSubstitute [latex]x=6[\/latex]&nbsp;and solve for [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\dfrac{25}{8}{\\left(6\\right)}^{3} \\\\[1mm] &amp;=675\\hfill \\end{align}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nThe graph of this equation is a simple cubic, as shown below.\n\n&nbsp;\n\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\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<strong>Do the graphs of all direct variation equations look like Example 1?<\/strong>\n\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>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nThe quantity [latex]y[\/latex]&nbsp;varies directly with the square of [latex]y[\/latex]. If [latex]y=24[\/latex]&nbsp;when [latex]x=3[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 4.\n\n[reveal-answer q=\"536994\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"536994\"]\n\n[latex]\\dfrac{128}{3}[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91391&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\">\n<\/iframe>\n\n<\/div>\nWatch this video to see a quick lesson in direct variation. &nbsp;You will see more worked examples.\n\nhttps:\/\/youtu.be\/plFOq4JaEyI\n<h2>Inverse and Joint Variation<\/h2>\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=\\Dfrac{14,000}{d}[\/latex] gives us the temperature in degrees Fahrenheit at a depth in feet below Earth\u2019s surface. Consider the Atlantic Ocean, which covers 22% of Earth\u2019s surface. At a certain location, at the depth of 500 feet, the temperature may be 28\u00b0F.\n\nIf we create a table&nbsp;we observe that, as the depth increases, the water temperature decreases.\n<table summary=\"..\">\n<thead>\n<tr>\n<th>d[\/latex], depth<\/th>\n<th>[latex]T=\\frac{\\text{14,000}}{d}[\/latex]<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>500 ft<\/td>\n<td>[latex]\\frac{14,000}{500}=28[\/latex]<\/td>\n<td>At a depth of 500 ft, the water temperature is 28\u00b0 F.<\/td>\n<\/tr>\n<tr>\n<td>350 ft<\/td>\n<td>[latex]\\frac{14,000}{350}=40[\/latex]<\/td>\n<td>At a depth of 350 ft, the water temperature is 40\u00b0 F.<\/td>\n<\/tr>\n<tr>\n<td>250 ft<\/td>\n<td>[latex]\\frac{14,000}{250}=56[\/latex]<\/td>\n<td>At a depth of 250 ft, the water temperature is 56\u00b0 F.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nWe notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be <strong>inversely proportional<\/strong> and each term <strong>varies inversely<\/strong> with the other. Inversely proportional relationships are also called <strong>inverse variations<\/strong>.\n\nFor our example, the graph&nbsp;depicts the <strong>inverse variation<\/strong>. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula [latex]y=\\dfrac{k}{x}[\/latex] for inverse variation in this case uses [latex]k=14,000[\/latex].\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222955\/CNX_Precalc_Figure_03_09_0032.jpg\" alt=\"Graph of y=(14000)\/x where the horizontal axis is labeled,\" width=\"487\" height=\"309\">\n<div class=\"textbox\">\n<h3>A General Note: Inverse Variation<\/h3>\nIf [latex]x[\/latex] and [latex]y[\/latex]&nbsp;are related by an equation of the form\n\n[latex]y=\\dfrac{k}{{x}^{n}}[\/latex]\n\nwhere [latex]k[\/latex]&nbsp;is a nonzero constant, then we say that [latex]y[\/latex]&nbsp;<strong>varies inversely<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex]. In <strong>inversely proportional<\/strong> relationships, or <strong>inverse variations<\/strong>, there is a constant multiple [latex]k={x}^{n}y[\/latex].\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Formula for an Inversely Proportional Relationship<\/h3>\nA tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.\n\n[reveal-answer q=\"81111\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"81111\"]\n\nRecall that multiplying speed by time gives distance. If we let [latex]t[\/latex]&nbsp;represent the drive time in hours, and [latex]v[\/latex]&nbsp;represent the velocity (speed or rate) at which the tourist drives, then [latex]vt=[\/latex]&nbsp;distance. Because the distance is fixed at 100 miles, [latex]vt=100[\/latex]. Solving this relationship for the time gives us our function.\n<p style=\"text-align: center;\">[latex]\\begin{align}t\\left(v\\right)&amp;=\\dfrac{100}{v} \\\\[1mm] &amp;=100{v}^{-1} \\end{align}[\/latex]<\/p>\nWe can see that the constant of variation is 100 and, although we can write the relationship using the negative exponent, it is more common to see it written as a fraction.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a description of an indirect variation problem, solve for an unknown.<strong>\n<\/strong><\/h3>\n<ol>\n \t<li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\n \t<li>Determine the constant of variation. You may need to multiply [latex]y[\/latex]&nbsp;by the specified power of [latex]x[\/latex]&nbsp;to determine the constant of variation.<\/li>\n \t<li>Use the constant of variation to write an equation for the relationship.<\/li>\n \t<li>Substitute known values into the equation to find the unknown.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Inverse Variation Problem<\/h3>\nA quantity [latex]y[\/latex]&nbsp;varies inversely with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]&nbsp;when [latex]x=2[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 6.\n\n[reveal-answer q=\"482072\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"482072\"]\n\nThe general formula for inverse variation with a cube is [latex]y=\\dfrac{k}{{x}^{3}}[\/latex]. The constant can be found by multiplying [latex]y[\/latex]&nbsp;by the cube of [latex]x[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align}k&amp;={x}^{3}y \\\\[1mm] &amp;={2}^{3}\\cdot 25 \\\\[1mm] &amp;=200 \\end{align}[\/latex]<\/p>\nNow we use the constant to write an equation that represents this relationship.\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\dfrac{k}{{x}^{3}},\\hspace{2mm}k=200 \\\\[1mm] y&amp;=\\dfrac{200}{{x}^{3}} \\end{align}[\/latex]<\/p>\nSubstitute [latex]x=6[\/latex]&nbsp;and solve for [latex]y[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align}y&amp;=\\dfrac{200}{{6}^{3}} \\\\[1mm] &amp;=\\dfrac{25}{27} \\end{align}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nThe graph of this equation is a rational function.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222957\/CNX_Precalc_Figure_03_09_0042.jpg\" alt=\"Graph of y=25\/(x^3) with the labeled points (2, 25) and (6, 25\/27).\" width=\"488\" height=\"292\">\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nA quantity [latex]y[\/latex]&nbsp;varies inversely with the square of [latex]x[\/latex]. If [latex]y=8[\/latex]&nbsp;when [latex]x=3[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 4.\n\n[reveal-answer q=\"285259\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"285259\"]\n\n[latex]\\dfrac{9}{2}[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91393&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"400\">\n<\/iframe>\n\n<\/div>\nThe following video presents a short lesson on inverse variation and includes more worked examples.\n\nhttps:\/\/youtu.be\/awp2vxqd-l4\n<h2>Joint Variation<\/h2>\nMany situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called <strong>joint variation<\/strong>. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable [latex]c[\/latex], cost, varies jointly with the number of students, [latex]n[\/latex], and the distance, [latex]d[\/latex].\n<div class=\"textbox\">\n<h3>A General Note: Joint Variation<\/h3>\nJoint variation occurs when a variable varies directly or inversely with multiple variables.\n\nFor instance, if [latex]x[\/latex]&nbsp;varies directly with both [latex]y[\/latex]&nbsp;and [latex]z[\/latex], we have [latex]x=kyz[\/latex]. If [latex]x[\/latex]&nbsp;varies directly with [latex]y[\/latex]&nbsp;and inversely with [latex]z[\/latex], we have [latex]x=\\dfrac{ky}{z}[\/latex]. Notice that we only use one constant in a joint variation equation.\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Problems Involving Joint Variation<\/h3>\nA quantity [latex]x[\/latex]&nbsp;varies directly with the square of [latex]y[\/latex]&nbsp;and inversely with the cube root of [latex]z[\/latex]. If [latex]x=6[\/latex]&nbsp;when [latex]y=2[\/latex]&nbsp;and [latex]z=8[\/latex], find [latex]x[\/latex]&nbsp;when [latex]y=1[\/latex]&nbsp;and [latex]z=27[\/latex].\n\n[reveal-answer q=\"396823\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"396823\"]\n\nBegin by writing an equation to show the relationship between the variables.\n<p style=\"text-align: center;\">[latex]x=\\dfrac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\nSubstitute [latex]x=6[\/latex], [latex]y=2[\/latex], and [latex]z=8[\/latex]&nbsp;to find the value of the constant [latex]k[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align}6&amp;=\\dfrac{k{2}^{2}}{\\sqrt[3]{8}} \\\\[1mm] 6&amp;=\\dfrac{4k}{2} \\\\[1mm] 3&amp;=k \\end{align}[\/latex]<\/p>\nNow we can substitute the value of the constant into the equation for the relationship.\n<p style=\"text-align: center;\">[latex]x=\\dfrac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\nTo find [latex]x[\/latex]&nbsp;when [latex]y=1[\/latex]&nbsp;and [latex]z=27[\/latex], we will substitute values for [latex]y[\/latex]&nbsp;and [latex]z[\/latex]&nbsp;into our equation.\n<p style=\"text-align: center;\">[latex]\\begin{align}x&amp;=\\dfrac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}} \\\\[1mm] &amp;=1 \\end{align}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[latex]x[\/latex] varies directly with the square of [latex]y[\/latex]&nbsp;and inversely with [latex]z[\/latex]. If [latex]x=40[\/latex]&nbsp;when [latex]y=4[\/latex]&nbsp;and [latex]z=2[\/latex], find [latex]x[\/latex]&nbsp;when [latex]y=10[\/latex]&nbsp;and [latex]z=25[\/latex].\n\n[reveal-answer q=\"286100\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"286100\"]\n\n[latex]x=20[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91394&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"400\">\n<\/iframe>\n\n<\/div>\n&nbsp;\n\nThe following video provides another worked example of a joint variation problem.\n\nhttps:\/\/youtu.be\/JREPATMScbM\n\n<section id=\"fs-id1165137898092\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\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=\\dfrac{k}{{x}^{n}},k\\text{ is a nonzero constant}\\\\[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section><section id=\"fs-id1165137419773\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165137723142\">\n \t<li>A relationship where one quantity is a constant multiplied by another quantity is called direct variation.<\/li>\n \t<li>Two variables that are directly proportional to one another will have a constant ratio.<\/li>\n \t<li>A relationship where one quantity is a constant divided by another quantity is called inverse variation.<\/li>\n \t<li>Two variables that are inversely proportional to one another will have a constant multiple.<\/li>\n \t<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 \t<dt><strong>constant of variation<\/strong><\/dt>\n \t<dd id=\"fs-id1165137735729\">the non-zero value [latex]k[\/latex] that helps define the relationship between variables in direct or inverse variation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137762202\" class=\"definition\">\n \t<dt><strong>direct variation<\/strong><\/dt>\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>\n<\/dl>\n<dl id=\"fs-id1165137462046\" class=\"definition\">\n \t<dt><strong>inverse variation<\/strong><\/dt>\n \t<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 \t<dt><strong>inversely proportional<\/strong><\/dt>\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>\n<\/dl>\n<dl id=\"fs-id1165137874546\" class=\"definition\">\n \t<dt><strong>joint variation<\/strong><\/dt>\n \t<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 \t<dt><strong>varies directly<\/strong><\/dt>\n \t<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 \t<dt><strong>varies inversely<\/strong><\/dt>\n \t<dd id=\"fs-id1165135439853\">a relationship where one quantity is a constant divided by the other quantity<\/dd>\n<\/dl>\n<\/section>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n<li class=\"li2\"><span class=\"s1\">Solve direct variation problems.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Solve inverse variation problems.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Solve problems involving joint variation.<\/span><\/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>Direct Variation<\/h2>\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].&nbsp;If we create a table, we observe that as the sales price increases, the earnings increase as well.<\/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<em>&nbsp;<\/em>= 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<em>&nbsp;<\/em>= 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<em>&nbsp;<\/em>= 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.<\/p>\n<p>The graph below&nbsp;represents 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]&nbsp;and [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>&nbsp;<\/em>and [latex]y[\/latex]&nbsp;are 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]&nbsp;<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]&nbsp;is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.<\/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]&nbsp;by the specified power of [latex]x[\/latex]&nbsp;to determine the constant of variation.<\/li>\n<li>Use the constant of variation to write an equation for the relationship.<\/li>\n<li>Substitute known values into the equation to find the unknown.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Direct Variation Problem<\/h3>\n<p>The quantity [latex]y[\/latex]&nbsp;varies directly with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]&nbsp;when [latex]x=2[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 6.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"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]&nbsp;by 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]&nbsp;and 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 Example 1?<\/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]&nbsp;varies directly with the square of [latex]y[\/latex]. If [latex]y=24[\/latex]&nbsp;when [latex]x=3[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 4.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"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><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91391&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<p>Watch this video to see a quick lesson in direct variation. &nbsp;You will see more worked examples.<\/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 and Joint 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=\\Dfrac{14,000}{d}[\/latex] gives us the temperature in degrees Fahrenheit at a depth in feet below Earth\u2019s surface. Consider the Atlantic Ocean, which covers 22% of Earth\u2019s surface. At a certain location, at the depth of 500 feet, the temperature may be 28\u00b0F.<\/p>\n<p>If we create a table&nbsp;we observe that, as the depth increases, the water temperature decreases.<\/p>\n<table summary=\"..\">\n<thead>\n<tr>\n<th>d[\/latex], depth<\/th>\n<th>[latex]T=\\frac{\\text{14,000}}{d}[\/latex]<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>500 ft<\/td>\n<td>[latex]\\frac{14,000}{500}=28[\/latex]<\/td>\n<td>At a depth of 500 ft, the water temperature is 28\u00b0 F.<\/td>\n<\/tr>\n<tr>\n<td>350 ft<\/td>\n<td>[latex]\\frac{14,000}{350}=40[\/latex]<\/td>\n<td>At a depth of 350 ft, the water temperature is 40\u00b0 F.<\/td>\n<\/tr>\n<tr>\n<td>250 ft<\/td>\n<td>[latex]\\frac{14,000}{250}=56[\/latex]<\/td>\n<td>At a depth of 250 ft, the water temperature is 56\u00b0 F.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be <strong>inversely proportional<\/strong> and each term <strong>varies inversely<\/strong> with the other. Inversely proportional relationships are also called <strong>inverse variations<\/strong>.<\/p>\n<p>For our example, the graph&nbsp;depicts the <strong>inverse variation<\/strong>. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula [latex]y=\\dfrac{k}{x}[\/latex] for inverse variation in this case uses [latex]k=14,000[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222955\/CNX_Precalc_Figure_03_09_0032.jpg\" alt=\"Graph of y=(14000)\/x where the horizontal axis is labeled,\" width=\"487\" height=\"309\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Inverse Variation<\/h3>\n<p>If [latex]x[\/latex] and [latex]y[\/latex]&nbsp;are related by an equation of the form<\/p>\n<p>[latex]y=\\dfrac{k}{{x}^{n}}[\/latex]<\/p>\n<p>where [latex]k[\/latex]&nbsp;is a nonzero constant, then we say that [latex]y[\/latex]&nbsp;<strong>varies inversely<\/strong> with the [latex]n[\/latex]th power of [latex]x[\/latex]. In <strong>inversely proportional<\/strong> relationships, or <strong>inverse variations<\/strong>, there is a constant multiple [latex]k={x}^{n}y[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Formula for an Inversely Proportional Relationship<\/h3>\n<p>A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q81111\">Show Solution<\/span><\/p>\n<div id=\"q81111\" class=\"hidden-answer\" style=\"display: none\">\n<p>Recall that multiplying speed by time gives distance. If we let [latex]t[\/latex]&nbsp;represent the drive time in hours, and [latex]v[\/latex]&nbsp;represent the velocity (speed or rate) at which the tourist drives, then [latex]vt=[\/latex]&nbsp;distance. Because the distance is fixed at 100 miles, [latex]vt=100[\/latex]. Solving this relationship for the time gives us our function.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}t\\left(v\\right)&=\\dfrac{100}{v} \\\\[1mm] &=100{v}^{-1} \\end{align}[\/latex]<\/p>\n<p>We can see that the constant of variation is 100 and, although we can write the relationship using the negative exponent, it is more common to see it written as a fraction.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a description of an indirect variation problem, solve for an unknown.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Identify the input, [latex]x[\/latex], and the output, [latex]y[\/latex].<\/li>\n<li>Determine the constant of variation. You may need to multiply [latex]y[\/latex]&nbsp;by the specified power of [latex]x[\/latex]&nbsp;to determine the constant of variation.<\/li>\n<li>Use the constant of variation to write an equation for the relationship.<\/li>\n<li>Substitute known values into the equation to find the unknown.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Inverse Variation Problem<\/h3>\n<p>A quantity [latex]y[\/latex]&nbsp;varies inversely with the cube of [latex]x[\/latex]. If [latex]y=25[\/latex]&nbsp;when [latex]x=2[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 6.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q482072\">Show Solution<\/span><\/p>\n<div id=\"q482072\" class=\"hidden-answer\" style=\"display: none\">\n<p>The general formula for inverse variation with a cube is [latex]y=\\dfrac{k}{{x}^{3}}[\/latex]. The constant can be found by multiplying [latex]y[\/latex]&nbsp;by the cube of [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}k&={x}^{3}y \\\\[1mm] &={2}^{3}\\cdot 25 \\\\[1mm] &=200 \\end{align}[\/latex]<\/p>\n<p>Now we use the constant to write an equation that represents this relationship.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&=\\dfrac{k}{{x}^{3}},\\hspace{2mm}k=200 \\\\[1mm] y&=\\dfrac{200}{{x}^{3}} \\end{align}[\/latex]<\/p>\n<p>Substitute [latex]x=6[\/latex]&nbsp;and solve for [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}y&=\\dfrac{200}{{6}^{3}} \\\\[1mm] &=\\dfrac{25}{27} \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph of this equation is a rational function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02222957\/CNX_Precalc_Figure_03_09_0042.jpg\" alt=\"Graph of y=25\/(x^3) with the labeled points (2, 25) and (6, 25\/27).\" width=\"488\" height=\"292\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A quantity [latex]y[\/latex]&nbsp;varies inversely with the square of [latex]x[\/latex]. If [latex]y=8[\/latex]&nbsp;when [latex]x=3[\/latex], find [latex]y[\/latex]&nbsp;when [latex]x[\/latex]&nbsp;is 4.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q285259\">Show Solution<\/span><\/p>\n<div id=\"q285259\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{9}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91393&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"400\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<p>The following video presents a short lesson on inverse variation and includes more worked examples.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-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]&nbsp;varies directly with both [latex]y[\/latex]&nbsp;and [latex]z[\/latex], we have [latex]x=kyz[\/latex]. If [latex]x[\/latex]&nbsp;varies directly with [latex]y[\/latex]&nbsp;and inversely with [latex]z[\/latex], we have [latex]x=\\dfrac{ky}{z}[\/latex]. Notice that we only use one constant in a joint variation equation.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Problems Involving Joint Variation<\/h3>\n<p>A quantity [latex]x[\/latex]&nbsp;varies directly with the square of [latex]y[\/latex]&nbsp;and inversely with the cube root of [latex]z[\/latex]. If [latex]x=6[\/latex]&nbsp;when [latex]y=2[\/latex]&nbsp;and [latex]z=8[\/latex], find [latex]x[\/latex]&nbsp;when [latex]y=1[\/latex]&nbsp;and [latex]z=27[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q396823\">Show Solution<\/span><\/p>\n<div id=\"q396823\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by writing an equation to show the relationship between the variables.<\/p>\n<p style=\"text-align: center;\">[latex]x=\\dfrac{k{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\n<p>Substitute [latex]x=6[\/latex], [latex]y=2[\/latex], and [latex]z=8[\/latex]&nbsp;to find the value of the constant [latex]k[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}6&=\\dfrac{k{2}^{2}}{\\sqrt[3]{8}} \\\\[1mm] 6&=\\dfrac{4k}{2} \\\\[1mm] 3&=k \\end{align}[\/latex]<\/p>\n<p>Now we can substitute the value of the constant into the equation for the relationship.<\/p>\n<p style=\"text-align: center;\">[latex]x=\\dfrac{3{y}^{2}}{\\sqrt[3]{z}}[\/latex]<\/p>\n<p>To find [latex]x[\/latex]&nbsp;when [latex]y=1[\/latex]&nbsp;and [latex]z=27[\/latex], we will substitute values for [latex]y[\/latex]&nbsp;and [latex]z[\/latex]&nbsp;into our equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x&=\\dfrac{3{\\left(1\\right)}^{2}}{\\sqrt[3]{27}} \\\\[1mm] &=1 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>[latex]x[\/latex] varies directly with the square of [latex]y[\/latex]&nbsp;and inversely with [latex]z[\/latex]. If [latex]x=40[\/latex]&nbsp;when [latex]y=4[\/latex]&nbsp;and [latex]z=2[\/latex], find [latex]x[\/latex]&nbsp;when [latex]y=10[\/latex]&nbsp;and [latex]z=25[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q286100\">Show Solution<\/span><\/p>\n<div id=\"q286100\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=20[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=91394&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"400\"><br \/>\n<\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The following video provides another worked example of a joint variation problem.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-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<section id=\"fs-id1165137898092\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\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=\\dfrac{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<h2>Key Concepts<\/h2>\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 [latex]k[\/latex] that 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-1876\">\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>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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>Question ID 91391, 91393, 91394. <strong>Authored by<\/strong>: Michael Jenck. <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>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><li>Inverse Variation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/awp2vxqd-l4\">https:\/\/youtu.be\/awp2vxqd-l4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Joint Variation: Determine the Variation Constant (Volume of a Cone). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Provided by<\/strong>: Joint Variation: Determine the Variation Constant (Volume of a Cone). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/JREPATMScbM\">https:\/\/youtu.be\/JREPATMScbM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":708740,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 91391, 91393, 91394\",\"author\":\"Michael Jenck\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Direct Variation Applications\",\"author\":\"James Sousa (Mathispower4u.com) 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