{"id":5711,"date":"2021-02-11T00:46:14","date_gmt":"2021-02-11T00:46:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/variation\/"},"modified":"2022-01-07T01:15:36","modified_gmt":"2022-01-07T01:15:36","slug":"3-6-direct-and-inverse-variation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/3-6-direct-and-inverse-variation\/","title":{"raw":"3.6 Direct and Inverse Variation","rendered":"3.6 Direct and Inverse Variation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul class=\"ul1\">\r\n \t<li class=\"li2\"><span class=\"s1\">Solve direct variation problems.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Solve inverse variation problems.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1165135356540\">A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section we will look at relationships, such as this one, between earnings, sales, and commission rate.<\/p>\r\n\r\n<h2>Direct Variation<\/h2>\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.\r\n<table style=\"height: 101px;\" summary=\"..\" width=\"722\">\r\n<thead>\r\n<tr>\r\n<th style=\"width: 166.5px;\">s, sales prices<\/th>\r\n<th style=\"width: 213.5px;\">e = 0.16s<\/th>\r\n<th style=\"width: 303.5px;\">Interpretation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 166.5px;\">$4,600<\/td>\r\n<td style=\"width: 213.5px;\">e = 0.16(4,600) = 736<\/td>\r\n<td style=\"width: 303.5px;\">A sale of a $4,600 vehicle results in $736 earnings.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 166.5px;\">$9,200<\/td>\r\n<td style=\"width: 213.5px;\">e = 0.16(9,200) = 1,472<\/td>\r\n<td style=\"width: 303.5px;\">A sale of a $9,200 vehicle results in $1472 earnings.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 166.5px;\">$18,400<\/td>\r\n<td style=\"width: 213.5px;\">e = 0.16(18,400) = 2,944<\/td>\r\n<td style=\"width: 303.5px;\">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.\r\n\r\nThe graph below 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}[\/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].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3741\/2019\/01\/16211148\/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] and [latex]y[\/latex] are related by an equation of the form\r\n<p style=\"text-align: center;\">[latex]y=k{x}[\/latex]<\/p>\r\nthen we say that the relationship is <strong>direct variation<\/strong> and [latex]y[\/latex] <strong>varies directly<\/strong> with [latex]x[\/latex]. In direct variation relationships, there is a nonzero constant ratio [latex]k=\\dfrac{y}{{x}}[\/latex], where [latex]k[\/latex] is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h1><strong>How to Solve a Direct Variation Problem<\/strong><\/h1>\r\n<strong>First<\/strong>: Find the <strong>constant of variation<\/strong>, by writing the model (the equation) for direct variation: [latex]y=k{x}[\/latex]\r\n\r\nPlug the given [latex]y[\/latex] and [latex]x[\/latex] into the model.\r\n\r\nSolve for [latex]k[\/latex], which is the <strong>constant of variation<\/strong>.\r\n\r\n<strong>Second<\/strong>: Write the <strong>equation of variation<\/strong> by plugging the constant of variation into the equation [latex]y=k{x}[\/latex].\r\n\r\n<strong>Third<\/strong>: Substitute the remaining [latex]x[\/latex] value into the equation of variation.\r\n\r\nSimplify to find [latex]y[\/latex].\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a Direct Variation Problem<\/h3>\r\nThe quantity [latex]y[\/latex] varies directly with [latex]x[\/latex]. If [latex]y=15[\/latex] when [latex]x=6[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 10.\r\n\r\n&nbsp;\r\n\r\n<strong>First<\/strong>: Find the <strong>constant of variation<\/strong>, by writing the model (the equation) for direct variation.\r\n<p style=\"text-align: center;\">[latex]y=k{x}[\/latex]<\/p>\r\nPlug [latex]y=15[\/latex] and [latex]x=6[\/latex] into the model.\r\n<p style=\"text-align: center;\">[latex]15=k{(6)}[\/latex]<\/p>\r\nSolve for [latex]k[\/latex] by dividing both sides of the equation by 6.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{15}{6}[\/latex] = [latex]\\dfrac{k(6)}{6}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]2.5=k{}[\/latex]<\/p>\r\nSo, the constant of variation is [latex]k=2.5[\/latex].\r\n\r\n&nbsp;\r\n\r\n<strong>Second<\/strong>: Write the <strong>equation of variation<\/strong> by plugging the constant of variation (in this case, [latex]k=2.5[\/latex] ) into the equation [latex]y=k{x}[\/latex].\r\n<p style=\"text-align: center;\">[latex]y=2.5{x}[\/latex]<\/p>\r\n<strong>Third<\/strong>: Substitute the remaining value (in this case, [latex]x=10[\/latex]) into the equation of variation.\r\n<p style=\"text-align: center;\">[latex]y=2.5{(10)}[\/latex]<\/p>\r\nSimplify to find [latex]y[\/latex].\r\n<p style=\"text-align: center;\">[latex]y=25[\/latex]<\/p>\r\nTherefore, if [latex]y[\/latex] varies directly with [latex]x[\/latex], and if [latex]y=15[\/latex] when [latex]x=6[\/latex], then [latex]y=25[\/latex] when [latex]x[\/latex] is 10.\r\n\r\n<\/div>\r\n<span id=\"MathJax-Element-15-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;y&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MJXc-Node-66\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-67\" class=\"mjx-mrow\"><span id=\"MJXc-Node-68\" class=\"mjx-mi\"> <\/span><\/span><\/span><\/span>Watch this video to see a quick lesson in direct variation. You will see more worked examples.\r\n\r\nhttps:\/\/youtu.be\/plFOq4JaEyI\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\n1) The quantity [latex]y[\/latex] varies directly with [latex]x[\/latex]. If [latex]y=16.8[\/latex] when [latex]x=2[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 3.\r\n<p style=\"padding-left: 30px;\">1a) Find the constant of variation.<\/p>\r\n[reveal-answer q=\"248951\"] Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"248951\"][latex]k=8.4[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">1b) Find the equation of variation.<\/p>\r\n[reveal-answer q=\"924726\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"924726\"][latex]y=8.4x[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">1c) Find y when [latex]x=3[\/latex].<\/p>\r\n[reveal-answer q=\"723146\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"723146\"][latex]y=25.2[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n2) The quantity [latex]y[\/latex] varies directly with [latex]x[\/latex]. If [latex]y=20.7[\/latex] when [latex]x=18[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 11.\r\n<p style=\"padding-left: 30px;\">2a) Find the constant of variation.<\/p>\r\n<p style=\"padding-left: 30px;\">[reveal-answer q=\"705555\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"705555\"][latex]k=1.15[\/latex][\/hidden-answer]<\/p>\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">2b) Find the equation of variation.<\/p>\r\n<p style=\"padding-left: 30px;\">[reveal-answer q=\"885303\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"885303\"][latex]y=1.15x[\/latex][\/hidden-answer]<\/p>\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">2c) Find y when [latex]x=11[\/latex].<\/p>\r\n<p style=\"padding-left: 30px;\">[reveal-answer q=\"79258\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"79258\"][latex]y=12.65[\/latex][\/hidden-answer]<\/p>\r\n&nbsp;\r\n\r\n3) The cost to fill your car\u2019s gas tank varies directly with the number of gallons you put in your tank. Let [latex]x[\/latex] be the number of gallons you put in the tank, and let [latex]y[\/latex] be the cost, in dollars, of the gasoline. Suppose that the car in front of you pumped 12 gallons of gas. You were able to see that the pump said $36.48. You are planning on pumping 15 gallons of gas into your car.\r\n<p style=\"padding-left: 30px;\">3a) Find the constant of variation for this scenario.<\/p>\r\n[reveal-answer q=\"907116\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"907116\"][latex]k=3.04[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">3b) What does that constant of variation represent in this scenario?<\/p>\r\n[reveal-answer q=\"938656\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"938656\"]The price per gallon of gasoline is $3.04.[\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">3c) Find the equation of variation for this scenario.<\/p>\r\n[reveal-answer q=\"103966\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"103966\"][latex]y=3.04x[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">3d) Find the amount you will pay for pumping 15 gallons of gas into your car.<\/p>\r\n[reveal-answer q=\"692006\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"692006\"]You will pay $45.60 for your 15 gallons of gas.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\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=\\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.\r\n\r\nIf we create a table we observe that, as the depth increases, the water temperature decreases.\r\n<table summary=\"..\">\r\n<thead>\r\n<tr>\r\n<th>d, 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 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].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3741\/2019\/01\/16211154\/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] are related by an equation of the form\r\n\r\n[latex]y=\\dfrac{k}{{x}}[\/latex]\r\n\r\nwhere [latex]k[\/latex] is a nonzero constant, then we say that [latex]y[\/latex] <strong>varies inversely<\/strong> with [latex]x[\/latex]. In <strong>inversely proportional<\/strong> relationships, or <strong>inverse variations<\/strong>, there is a constant multiple [latex]k={x}y[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h1><strong>How to Solve an Inverse Variation Problem<\/strong><\/h1>\r\n<strong>First<\/strong>: Find the <strong>constant of variation<\/strong>, by writing the model (the equation) for inverse variation: [latex]y=\\dfrac{k}{{x}}[\/latex]\r\n\r\nPlug the given [latex]y[\/latex] and [latex]x[\/latex] into the model.\r\n\r\nSolve for [latex]k[\/latex], which is the <strong>constant of variation<\/strong>.\r\n\r\n<strong>Second<\/strong>: Write the <strong>equation of variation<\/strong> by plugging the constant of variation into the equation [latex]y=\\dfrac{k}{{x}}[\/latex].\r\n\r\n<strong>Third<\/strong>: Substitute the remaining [latex]x[\/latex] value into the equation of variation.\r\n\r\nSimplify to find [latex]y[\/latex].\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<div class=\"textbox exercises\">\r\n<h3>Example: Solving an Inverse Variation Problem<\/h3>\r\nA quantity [latex]y[\/latex] varies inversely with [latex]x[\/latex]. If [latex]y=4.65[\/latex] when [latex]x=2[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 5.\r\n\r\n<strong>First<\/strong>: Find the <strong>constant of variation<\/strong>, by writing the model (the equation) for inverse variation.\r\n<p style=\"text-align: center;\">[latex]y=\\dfrac{k}{{x}}[\/latex]<\/p>\r\nPlug [latex]y=4.65[\/latex] and [latex]x=2[\/latex] into the model.\r\n<p style=\"text-align: center;\">[latex]4.65=\\dfrac{k}{2}[\/latex]<\/p>\r\nSolve for [latex]k[\/latex] by multiplying both sides of the equation by 2.\r\n<p style=\"text-align: center;\">[latex]4.65x2=\\dfrac{k}{2}[\/latex]x2<\/p>\r\n<p style=\"text-align: center;\">[latex]9.3=k{}[\/latex]<\/p>\r\nSo, the constant of variation is [latex]k=9.3[\/latex].\r\n\r\n&nbsp;\r\n\r\n<strong>Second<\/strong>: Write the <strong>equation of variation<\/strong> by plugging the constant of variation (in this case, [latex]k=9.3[\/latex] ) into the equation [latex]y=\\dfrac{k}{{x}}[\/latex]\r\n<p style=\"text-align: center;\">[latex]y=\\dfrac{9.3}{{x}}[\/latex]<\/p>\r\n<strong>Third<\/strong>: Substitute the remaining value (in this case, [latex]x=5[\/latex]) into the equation of variation.\r\n<p style=\"text-align: center;\">[latex]y=\\dfrac{9.3}{{5}}[\/latex]<\/p>\r\nSimplify to find [latex]y[\/latex].\r\n<p style=\"text-align: center;\">[latex]y=1.86[\/latex]<\/p>\r\nTherefore, if [latex]y[\/latex] varies inversely with [latex]x[\/latex], and [latex]y=4.65[\/latex] when [latex]x=2[\/latex], then [latex]y=1.86[\/latex] when [latex]x[\/latex] is 5.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n&nbsp;\r\n\r\n4) The quantity [latex]y[\/latex] varies inversely with [latex]x[\/latex]. If [latex]y=4.8[\/latex] when [latex]x=55[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 8.\r\n<p style=\"padding-left: 30px;\">4a) Find the constant of variation.<\/p>\r\n[reveal-answer q=\"691883\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"691883\"][latex]k=264[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">4b) Find the equation of variation.<\/p>\r\n[reveal-answer q=\"77582\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"77582\"][latex]y=\\dfrac{264}{x}[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">4c) Find y when [latex]x=8[\/latex].<\/p>\r\n[reveal-answer q=\"787388\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"787388\"][latex]y=33[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n5) The quantity [latex]y[\/latex] varies inversely with [latex]x[\/latex]. If [latex]y=11.25[\/latex] when [latex]x=4[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 6.\r\n<p style=\"padding-left: 30px;\">5a) Find the constant of variation.<\/p>\r\n[reveal-answer q=\"842776\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"842776\"][latex]k=45[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">5b) Find the equation of variation.<\/p>\r\n[reveal-answer q=\"936380\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"936380\"][latex]y=\\dfrac{45}{x}[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">5c) Find y when [latex]x=6[\/latex].<\/p>\r\n[reveal-answer q=\"276223\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"276223\"][latex]y=7.5[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n6) The time it takes to put up a fence varies inversely with the number of people working on the fence. Let [latex]x[\/latex] be the number of people working on the fence, and let [latex]y[\/latex] time it takes to complete the fence, in hours. Suppose your neighbor's fence had 5 people working on it and it took 14 hours. You know that you will have 4 people to work on your fence of the same length. You are wondering how long it will take to complete your fence.\r\n<p style=\"padding-left: 30px;\">6a) Find the constant of variation for this scenario.<\/p>\r\n[reveal-answer q=\"244851\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"244851\"][latex]k=70[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">6b) What does that constant of variation represent in this scenario?<\/p>\r\n[reveal-answer q=\"238471\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"238471\"]If one person were to put up the fence (alone), it would take that person 70 hours. These are called \"people-hours\". That is the total amount of time to complete the job.[\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">6c) Find the equation of variation for this scenario.<\/p>\r\n[reveal-answer q=\"730307\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"730307\"][latex]y=\\dfrac{70}{x}[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n<p style=\"padding-left: 30px;\">6d) Find the amount of time that it will take 4 people to finish your fence.<\/p>\r\n[reveal-answer q=\"385423\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"385423\"]It will take 17.5 hours for 4 people to finish my fence.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<section id=\"fs-id1165137898092\" class=\"key-equations\"><\/section><section id=\"fs-id1165137419773\" class=\"key-concepts\">\r\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>\r\n\r\n<\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/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<\/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]. If we create a table, we observe that as the sales price increases, the earnings increase as well.<\/p>\n<table style=\"height: 101px; width: 722px;\" summary=\"..\">\n<thead>\n<tr>\n<th style=\"width: 166.5px;\">s, sales prices<\/th>\n<th style=\"width: 213.5px;\">e = 0.16s<\/th>\n<th style=\"width: 303.5px;\">Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 166.5px;\">$4,600<\/td>\n<td style=\"width: 213.5px;\">e = 0.16(4,600) = 736<\/td>\n<td style=\"width: 303.5px;\">A sale of a $4,600 vehicle results in $736 earnings.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 166.5px;\">$9,200<\/td>\n<td style=\"width: 213.5px;\">e = 0.16(9,200) = 1,472<\/td>\n<td style=\"width: 303.5px;\">A sale of a $9,200 vehicle results in $1472 earnings.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 166.5px;\">$18,400<\/td>\n<td style=\"width: 213.5px;\">e = 0.16(18,400) = 2,944<\/td>\n<td style=\"width: 303.5px;\">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 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}[\/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].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3741\/2019\/01\/16211148\/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] and [latex]y[\/latex] are related by an equation of the form<\/p>\n<p style=\"text-align: center;\">[latex]y=k{x}[\/latex]<\/p>\n<p>then we say that the relationship is <strong>direct variation<\/strong> and [latex]y[\/latex] <strong>varies directly<\/strong> with [latex]x[\/latex]. In direct variation relationships, there is a nonzero constant ratio [latex]k=\\dfrac{y}{{x}}[\/latex], where [latex]k[\/latex] is called the <strong>constant of variation<\/strong>, which help defines the relationship between the variables.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h1><strong>How to Solve a Direct Variation Problem<\/strong><\/h1>\n<p><strong>First<\/strong>: Find the <strong>constant of variation<\/strong>, by writing the model (the equation) for direct variation: [latex]y=k{x}[\/latex]<\/p>\n<p>Plug the given [latex]y[\/latex] and [latex]x[\/latex] into the model.<\/p>\n<p>Solve for [latex]k[\/latex], which is the <strong>constant of variation<\/strong>.<\/p>\n<p><strong>Second<\/strong>: Write the <strong>equation of variation<\/strong> by plugging the constant of variation into the equation [latex]y=k{x}[\/latex].<\/p>\n<p><strong>Third<\/strong>: Substitute the remaining [latex]x[\/latex] value into the equation of variation.<\/p>\n<p>Simplify to find [latex]y[\/latex].<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Direct Variation Problem<\/h3>\n<p>The quantity [latex]y[\/latex] varies directly with [latex]x[\/latex]. If [latex]y=15[\/latex] when [latex]x=6[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 10.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>First<\/strong>: Find the <strong>constant of variation<\/strong>, by writing the model (the equation) for direct variation.<\/p>\n<p style=\"text-align: center;\">[latex]y=k{x}[\/latex]<\/p>\n<p>Plug [latex]y=15[\/latex] and [latex]x=6[\/latex] into the model.<\/p>\n<p style=\"text-align: center;\">[latex]15=k{(6)}[\/latex]<\/p>\n<p>Solve for [latex]k[\/latex] by dividing both sides of the equation by 6.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{15}{6}[\/latex] = [latex]\\dfrac{k(6)}{6}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]2.5=k{}[\/latex]<\/p>\n<p>So, the constant of variation is [latex]k=2.5[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Second<\/strong>: Write the <strong>equation of variation<\/strong> by plugging the constant of variation (in this case, [latex]k=2.5[\/latex] ) into the equation [latex]y=k{x}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]y=2.5{x}[\/latex]<\/p>\n<p><strong>Third<\/strong>: Substitute the remaining value (in this case, [latex]x=10[\/latex]) into the equation of variation.<\/p>\n<p style=\"text-align: center;\">[latex]y=2.5{(10)}[\/latex]<\/p>\n<p>Simplify to find [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]y=25[\/latex]<\/p>\n<p>Therefore, if [latex]y[\/latex] varies directly with [latex]x[\/latex], and if [latex]y=15[\/latex] when [latex]x=6[\/latex], then [latex]y=25[\/latex] when [latex]x[\/latex] is 10.<\/p>\n<\/div>\n<p><span id=\"MathJax-Element-15-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;y&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MJXc-Node-66\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-67\" class=\"mjx-mrow\"><span id=\"MJXc-Node-68\" class=\"mjx-mi\"> <\/span><\/span><\/span><\/span>Watch this video to see a quick lesson in direct variation. 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<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p>1) The quantity [latex]y[\/latex] varies directly with [latex]x[\/latex]. If [latex]y=16.8[\/latex] when [latex]x=2[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 3.<\/p>\n<p style=\"padding-left: 30px;\">1a) Find the constant of variation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q248951\"> Show Answer<\/span><\/p>\n<div id=\"q248951\" class=\"hidden-answer\" style=\"display: none\">[latex]k=8.4[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">1b) Find the equation of variation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q924726\">Show Answer<\/span><\/p>\n<div id=\"q924726\" class=\"hidden-answer\" style=\"display: none\">[latex]y=8.4x[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">1c) Find y when [latex]x=3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q723146\">Show Answer<\/span><\/p>\n<div id=\"q723146\" class=\"hidden-answer\" style=\"display: none\">[latex]y=25.2[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>2) The quantity [latex]y[\/latex] varies directly with [latex]x[\/latex]. If [latex]y=20.7[\/latex] when [latex]x=18[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 11.<\/p>\n<p style=\"padding-left: 30px;\">2a) Find the constant of variation.<\/p>\n<p style=\"padding-left: 30px;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q705555\">Show Answer<\/span><\/p>\n<div id=\"q705555\" class=\"hidden-answer\" style=\"display: none\">[latex]k=1.15[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">2b) Find the equation of variation.<\/p>\n<p style=\"padding-left: 30px;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q885303\">Show Answer<\/span><\/p>\n<div id=\"q885303\" class=\"hidden-answer\" style=\"display: none\">[latex]y=1.15x[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">2c) Find y when [latex]x=11[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q79258\">Show Answer<\/span><\/p>\n<div id=\"q79258\" class=\"hidden-answer\" style=\"display: none\">[latex]y=12.65[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>3) The cost to fill your car\u2019s gas tank varies directly with the number of gallons you put in your tank. Let [latex]x[\/latex] be the number of gallons you put in the tank, and let [latex]y[\/latex] be the cost, in dollars, of the gasoline. Suppose that the car in front of you pumped 12 gallons of gas. You were able to see that the pump said $36.48. You are planning on pumping 15 gallons of gas into your car.<\/p>\n<p style=\"padding-left: 30px;\">3a) Find the constant of variation for this scenario.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q907116\">Show Answer<\/span><\/p>\n<div id=\"q907116\" class=\"hidden-answer\" style=\"display: none\">[latex]k=3.04[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">3b) What does that constant of variation represent in this scenario?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q938656\">Show Answer<\/span><\/p>\n<div id=\"q938656\" class=\"hidden-answer\" style=\"display: none\">The price per gallon of gasoline is $3.04.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">3c) Find the equation of variation for this scenario.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q103966\">Show Answer<\/span><\/p>\n<div id=\"q103966\" class=\"hidden-answer\" style=\"display: none\">[latex]y=3.04x[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">3d) Find the amount you will pay for pumping 15 gallons of gas into your car.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q692006\">Show Answer<\/span><\/p>\n<div id=\"q692006\" class=\"hidden-answer\" style=\"display: none\">You will pay $45.60 for your 15 gallons of gas.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\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=\\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 we observe that, as the depth increases, the water temperature decreases.<\/p>\n<table summary=\"..\">\n<thead>\n<tr>\n<th>d, 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 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\/3741\/2019\/01\/16211154\/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] are related by an equation of the form<\/p>\n<p>[latex]y=\\dfrac{k}{{x}}[\/latex]<\/p>\n<p>where [latex]k[\/latex] is a nonzero constant, then we say that [latex]y[\/latex] <strong>varies inversely<\/strong> with [latex]x[\/latex]. In <strong>inversely proportional<\/strong> relationships, or <strong>inverse variations<\/strong>, there is a constant multiple [latex]k={x}y[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h1><strong>How to Solve an Inverse Variation Problem<\/strong><\/h1>\n<p><strong>First<\/strong>: Find the <strong>constant of variation<\/strong>, by writing the model (the equation) for inverse variation: [latex]y=\\dfrac{k}{{x}}[\/latex]<\/p>\n<p>Plug the given [latex]y[\/latex] and [latex]x[\/latex] into the model.<\/p>\n<p>Solve for [latex]k[\/latex], which is the <strong>constant of variation<\/strong>.<\/p>\n<p><strong>Second<\/strong>: Write the <strong>equation of variation<\/strong> by plugging the constant of variation into the equation [latex]y=\\dfrac{k}{{x}}[\/latex].<\/p>\n<p><strong>Third<\/strong>: Substitute the remaining [latex]x[\/latex] value into the equation of variation.<\/p>\n<p>Simplify to find [latex]y[\/latex].<\/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<div class=\"textbox exercises\">\n<h3>Example: Solving an Inverse Variation Problem<\/h3>\n<p>A quantity [latex]y[\/latex] varies inversely with [latex]x[\/latex]. If [latex]y=4.65[\/latex] when [latex]x=2[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 5.<\/p>\n<p><strong>First<\/strong>: Find the <strong>constant of variation<\/strong>, by writing the model (the equation) for inverse variation.<\/p>\n<p style=\"text-align: center;\">[latex]y=\\dfrac{k}{{x}}[\/latex]<\/p>\n<p>Plug [latex]y=4.65[\/latex] and [latex]x=2[\/latex] into the model.<\/p>\n<p style=\"text-align: center;\">[latex]4.65=\\dfrac{k}{2}[\/latex]<\/p>\n<p>Solve for [latex]k[\/latex] by multiplying both sides of the equation by 2.<\/p>\n<p style=\"text-align: center;\">[latex]4.65x2=\\dfrac{k}{2}[\/latex]x2<\/p>\n<p style=\"text-align: center;\">[latex]9.3=k{}[\/latex]<\/p>\n<p>So, the constant of variation is [latex]k=9.3[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Second<\/strong>: Write the <strong>equation of variation<\/strong> by plugging the constant of variation (in this case, [latex]k=9.3[\/latex] ) into the equation [latex]y=\\dfrac{k}{{x}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]y=\\dfrac{9.3}{{x}}[\/latex]<\/p>\n<p><strong>Third<\/strong>: Substitute the remaining value (in this case, [latex]x=5[\/latex]) into the equation of variation.<\/p>\n<p style=\"text-align: center;\">[latex]y=\\dfrac{9.3}{{5}}[\/latex]<\/p>\n<p>Simplify to find [latex]y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]y=1.86[\/latex]<\/p>\n<p>Therefore, if [latex]y[\/latex] varies inversely with [latex]x[\/latex], and [latex]y=4.65[\/latex] when [latex]x=2[\/latex], then [latex]y=1.86[\/latex] when [latex]x[\/latex] is 5.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>&nbsp;<\/p>\n<p>4) The quantity [latex]y[\/latex] varies inversely with [latex]x[\/latex]. If [latex]y=4.8[\/latex] when [latex]x=55[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 8.<\/p>\n<p style=\"padding-left: 30px;\">4a) Find the constant of variation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q691883\">Show Answer<\/span><\/p>\n<div id=\"q691883\" class=\"hidden-answer\" style=\"display: none\">[latex]k=264[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">4b) Find the equation of variation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q77582\">Show Answer<\/span><\/p>\n<div id=\"q77582\" class=\"hidden-answer\" style=\"display: none\">[latex]y=\\dfrac{264}{x}[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">4c) Find y when [latex]x=8[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q787388\">Show Answer<\/span><\/p>\n<div id=\"q787388\" class=\"hidden-answer\" style=\"display: none\">[latex]y=33[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>5) The quantity [latex]y[\/latex] varies inversely with [latex]x[\/latex]. If [latex]y=11.25[\/latex] when [latex]x=4[\/latex], find [latex]y[\/latex] when [latex]x[\/latex] is 6.<\/p>\n<p style=\"padding-left: 30px;\">5a) Find the constant of variation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q842776\">Show Answer<\/span><\/p>\n<div id=\"q842776\" class=\"hidden-answer\" style=\"display: none\">[latex]k=45[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">5b) Find the equation of variation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q936380\">Show Answer<\/span><\/p>\n<div id=\"q936380\" class=\"hidden-answer\" style=\"display: none\">[latex]y=\\dfrac{45}{x}[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">5c) Find y when [latex]x=6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q276223\">Show Answer<\/span><\/p>\n<div id=\"q276223\" class=\"hidden-answer\" style=\"display: none\">[latex]y=7.5[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>6) The time it takes to put up a fence varies inversely with the number of people working on the fence. Let [latex]x[\/latex] be the number of people working on the fence, and let [latex]y[\/latex] time it takes to complete the fence, in hours. Suppose your neighbor&#8217;s fence had 5 people working on it and it took 14 hours. You know that you will have 4 people to work on your fence of the same length. You are wondering how long it will take to complete your fence.<\/p>\n<p style=\"padding-left: 30px;\">6a) Find the constant of variation for this scenario.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q244851\">Show Answer<\/span><\/p>\n<div id=\"q244851\" class=\"hidden-answer\" style=\"display: none\">[latex]k=70[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">6b) What does that constant of variation represent in this scenario?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q238471\">Show Answer<\/span><\/p>\n<div id=\"q238471\" class=\"hidden-answer\" style=\"display: none\">If one person were to put up the fence (alone), it would take that person 70 hours. These are called &#8220;people-hours&#8221;. That is the total amount of time to complete the job.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">6c) Find the equation of variation for this scenario.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q730307\">Show Answer<\/span><\/p>\n<div id=\"q730307\" class=\"hidden-answer\" style=\"display: none\">[latex]y=\\dfrac{70}{x}[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 30px;\">6d) Find the amount of time that it will take 4 people to finish your fence.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q385423\">Show Answer<\/span><\/p>\n<div id=\"q385423\" class=\"hidden-answer\" style=\"display: none\">It will take 17.5 hours for 4 people to finish my fence.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<section id=\"fs-id1165137898092\" class=\"key-equations\"><\/section>\n<section id=\"fs-id1165137419773\" class=\"key-concepts\">\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>\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-5711\">\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><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":23485,"menu_order":7,"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 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