{"id":826,"date":"2025-06-20T17:15:57","date_gmt":"2025-06-20T17:15:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=826"},"modified":"2025-09-09T18:57:20","modified_gmt":"2025-09-09T18:57:20","slug":"separation-of-variables-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/separation-of-variables-fresh-take\/","title":{"raw":"Separation of Variables: Fresh Take","rendered":"Separation of Variables: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve differential equations by separating variables<\/li>\r\n \t<li>Apply separation of variables to real-world problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Separation of Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Think of separation of variables like untangling a knot\u2014you separate the [latex]x[\/latex] stuff from the [latex]y[\/latex] stuff, then deal with each side independently.<\/p>\r\n<p class=\"whitespace-normal break-words\">When Can You Use This Method? Your equation must be <strong>separable<\/strong>: [latex]y' = f(x) \\cdot g(y)[\/latex]. The right side breaks into one function of [latex]x[\/latex] times one function of [latex]y[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Quick Recognition Test:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]y' = (x^2 - 4)(3y + 2)[\/latex] \u2192 <strong>Separable<\/strong> \u2713<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y' = xy + 2x[\/latex] \u2192 <strong>Separable<\/strong> (factor as [latex]x(y + 2)[\/latex]) \u2713<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y' = x + y^2[\/latex] \u2192 <strong>Not separable<\/strong> (can't factor) \u2717<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Check for constant solutions:<\/strong> Set [latex]g(y) = 0[\/latex] to find equilibrium points<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Separate variables:<\/strong> Rearrange to [latex]\\frac{dy}{g(y)} = f(x)dx[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Integrate both sides:<\/strong> [latex]\\int \\frac{dy}{g(y)} = \\int f(x)dx[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Solve for [latex]y[\/latex]<\/strong> (if possible\u2014sometimes you're stuck with implicit solutions)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Apply initial conditions<\/strong> to find specific constants<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">You're essentially moving all the [latex]y[\/latex] terms to one side and all the [latex]x[\/latex] terms to the other, then integrating each side separately.<\/p>\r\n<p class=\"whitespace-normal break-words\">Don't forget those constant solutions from Step 1! They're often missed but represent important equilibrium states.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1170571123151\" data-type=\"problem\">\r\n<p id=\"fs-id1170573402078\">Use the method of separation of variables to find a general solution to the differential equation [latex]y^{\\prime} =2xy+3y - 4x - 6[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558897\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558897\"]\r\n<div id=\"fs-id1170573255391\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1170573328131\">First factor the right-hand side of the equation by grouping, then use the five-step strategy of separation of variables.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558898\"]\r\n<div id=\"fs-id1170573388976\" data-type=\"solution\">\r\n<p id=\"fs-id1170571375609\">[latex]y=2+C{e}^{{x}^{2}+3x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1170571300752\" data-type=\"problem\">\r\n<p id=\"fs-id1170571097577\">Find the solution to the initial-value problem<\/p>\r\n\r\n<div id=\"fs-id1170573582993\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]6y^{\\prime} =\\left(2x+1\\right)\\left({y}^{2}-2y - 8\\right),y\\left(0\\right)=-3[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170573577573\">using the method of separation of variables.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558894\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558894\"]\r\n<div id=\"fs-id1170571228168\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1170571124616\">Follow the steps for separation of variables to solve the initial-value problem.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558895\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558895\"]\r\n<div id=\"fs-id1170573288989\" data-type=\"solution\">\r\n<p id=\"fs-id1170571136712\">[latex]y=\\frac{4+14{e}^{{x}^{2}+x}}{1 - 7{e}^{{x}^{2}+x}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Applications of Separation of Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Separation of variables isn't just an abstract math technique\u2014it's the key to solving problems you encounter in labs, kitchens, and anywhere things mix, flow, or change temperature.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Two Big Categories:<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Tank\/Mixing Problems:<\/strong> Track how concentrations change over time<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Setup:<\/strong> [latex]\\frac{du}{dt} = \\text{INFLOW RATE} - \\text{OUTFLOW RATE}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Key insight:<\/strong> Inflow brings in new stuff, outflow removes mixed stuff<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Common pattern:<\/strong> Salt solution flowing into a tank while mixed solution flows out<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Cooling\/Heating Problems:<\/strong> Newton's Law of Cooling<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Setup:<\/strong> [latex]\\frac{dT}{dt} = k(T - T_s)[\/latex] where [latex]T_s[\/latex] is ambient temperature<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Key insight:<\/strong> Temperature change rate depends on the temperature difference<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Common pattern:<\/strong> Hot coffee cooling to room temperature, pizza cooling after leaving the oven<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Step 1:<\/strong> Identify what's changing (salt amount, temperature, etc.)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Step 2:<\/strong> Set up the rate equation based on physical principles<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Step 3:<\/strong> Apply separation of variables<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Step 4:<\/strong> Use given information to find constants<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Step 5:<\/strong> Answer the specific question asked<\/li>\r\n<\/ul>\r\nUnits matter! Make sure rates, concentrations, and times all work together.\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1170571290129\" data-type=\"problem\">\r\n<p id=\"fs-id1170571290132\">A tank contains [latex]3[\/latex] kilograms of salt dissolved in [latex]75[\/latex] liters of water. A salt solution of [latex]0.4\\text{kg salt\/L}[\/latex] is pumped into the tank at a rate of [latex]6\\text{L\/min}[\/latex] and is drained at the same rate. Solve for the salt concentration at time [latex]t[\/latex]. Assume the tank is well mixed at all times.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558891\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558891\"]\r\n<div id=\"fs-id1170573356102\" data-type=\"commentary\" data-element-type=\"hint\">\r\n\r\nFollow the steps in the example:\u00a0Determining Salt Concentration over Time and determine an expression for INFLOW and OUTFLOW. Formulate an initial-value problem, and then solve it.\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558892\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558892\"]\r\n<div id=\"fs-id1170571096071\" data-type=\"solution\">\r\n<p id=\"fs-id1170571096073\">Initial value problem:<\/p>\r\n<p id=\"fs-id1170571096076\">[latex]\\frac{du}{dt}=2.4-\\frac{2u}{25},u\\left(0\\right)=3[\/latex]<\/p>\r\n<p id=\"fs-id1170571145647\">[latex]\\text{Solution:}u\\left(t\\right)=30 - 27{e}^{\\frac{\\text{-}t}{50}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1170571110826\" data-type=\"problem\">\r\n<p id=\"fs-id1170571110828\">A cake is removed from the oven after baking thoroughly, and the temperature of the oven is [latex]450^\\circ\\text{F}\\text{.}[\/latex] The temperature of the kitchen is [latex]70^\\circ\\text{F}[\/latex], and after [latex]10[\/latex] minutes the temperature of the cake is [latex]430^\\circ\\text{F}\\text{.}[\/latex]<\/p>\r\n\r\n<ol id=\"fs-id1170571143291\" type=\"a\">\r\n \t<li>Write the appropriate initial-value problem to describe this situation.<\/li>\r\n \t<li>Solve the initial-value problem for [latex]T\\left(t\\right)[\/latex].<\/li>\r\n \t<li>How long will it take until the temperature of the cake is within [latex]5^\\circ\\text{F}[\/latex] of room temperature?<\/li>\r\n<\/ol>\r\n<\/div>\r\n[reveal-answer q=\"44558879\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558879\"]\r\n<div id=\"fs-id1170571137906\" data-type=\"commentary\" data-element-type=\"hint\">\r\n\r\nDetermine the values of [latex]{T}_{s}[\/latex] and [latex]{T}_{0}[\/latex] then use the example:\u00a0Solving an Initial-Value Problem.\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558889\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558889\"]\r\n<div id=\"fs-id1170571292457\" data-type=\"solution\">\r\n<ol id=\"fs-id1170571292459\" type=\"a\">\r\n \t<li style=\"text-align: left;\">Initial-value problem<span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{dT}{dt}=k\\left(T - 70\\right),T\\left(0\\right)=450[\/latex]<\/li>\r\n \t<li>[latex]T\\left(t\\right)=70+380{e}^{kt}[\/latex]<\/li>\r\n \t<li>Approximately [latex]114[\/latex] minutes.<\/li>\r\n<\/ol>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve differential equations by separating variables<\/li>\n<li>Apply separation of variables to real-world problems<\/li>\n<\/ul>\n<\/section>\n<h2>Separation of Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Think of separation of variables like untangling a knot\u2014you separate the [latex]x[\/latex] stuff from the [latex]y[\/latex] stuff, then deal with each side independently.<\/p>\n<p class=\"whitespace-normal break-words\">When Can You Use This Method? Your equation must be <strong>separable<\/strong>: [latex]y' = f(x) \\cdot g(y)[\/latex]. The right side breaks into one function of [latex]x[\/latex] times one function of [latex]y[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Quick Recognition Test:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]y' = (x^2 - 4)(3y + 2)[\/latex] \u2192 <strong>Separable<\/strong> \u2713<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y' = xy + 2x[\/latex] \u2192 <strong>Separable<\/strong> (factor as [latex]x(y + 2)[\/latex]) \u2713<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y' = x + y^2[\/latex] \u2192 <strong>Not separable<\/strong> (can&#8217;t factor) \u2717<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Check for constant solutions:<\/strong> Set [latex]g(y) = 0[\/latex] to find equilibrium points<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Separate variables:<\/strong> Rearrange to [latex]\\frac{dy}{g(y)} = f(x)dx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Integrate both sides:<\/strong> [latex]\\int \\frac{dy}{g(y)} = \\int f(x)dx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Solve for [latex]y[\/latex]<\/strong> (if possible\u2014sometimes you&#8217;re stuck with implicit solutions)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Apply initial conditions<\/strong> to find specific constants<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">You&#8217;re essentially moving all the [latex]y[\/latex] terms to one side and all the [latex]x[\/latex] terms to the other, then integrating each side separately.<\/p>\n<p class=\"whitespace-normal break-words\">Don&#8217;t forget those constant solutions from Step 1! They&#8217;re often missed but represent important equilibrium states.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1170571123151\" data-type=\"problem\">\n<p id=\"fs-id1170573402078\">Use the method of separation of variables to find a general solution to the differential equation [latex]y^{\\prime} =2xy+3y - 4x - 6[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558897\">Hint<\/button><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573255391\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1170573328131\">First factor the right-hand side of the equation by grouping, then use the five-step strategy of separation of variables.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558898\">Show Solution<\/button><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573388976\" data-type=\"solution\">\n<p id=\"fs-id1170571375609\">[latex]y=2+C{e}^{{x}^{2}+3x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1170571300752\" data-type=\"problem\">\n<p id=\"fs-id1170571097577\">Find the solution to the initial-value problem<\/p>\n<div id=\"fs-id1170573582993\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]6y^{\\prime} =\\left(2x+1\\right)\\left({y}^{2}-2y - 8\\right),y\\left(0\\right)=-3[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170573577573\">using the method of separation of variables.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558894\">Hint<\/button><\/p>\n<div id=\"q44558894\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170571228168\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1170571124616\">Follow the steps for separation of variables to solve the initial-value problem.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558895\">Show Solution<\/button><\/p>\n<div id=\"q44558895\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573288989\" data-type=\"solution\">\n<p id=\"fs-id1170571136712\">[latex]y=\\frac{4+14{e}^{{x}^{2}+x}}{1 - 7{e}^{{x}^{2}+x}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Applications of Separation of Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Separation of variables isn&#8217;t just an abstract math technique\u2014it&#8217;s the key to solving problems you encounter in labs, kitchens, and anywhere things mix, flow, or change temperature.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Two Big Categories:<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Tank\/Mixing Problems:<\/strong> Track how concentrations change over time<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Setup:<\/strong> [latex]\\frac{du}{dt} = \\text{INFLOW RATE} - \\text{OUTFLOW RATE}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Key insight:<\/strong> Inflow brings in new stuff, outflow removes mixed stuff<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Common pattern:<\/strong> Salt solution flowing into a tank while mixed solution flows out<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Cooling\/Heating Problems:<\/strong> Newton&#8217;s Law of Cooling<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Setup:<\/strong> [latex]\\frac{dT}{dt} = k(T - T_s)[\/latex] where [latex]T_s[\/latex] is ambient temperature<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Key insight:<\/strong> Temperature change rate depends on the temperature difference<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Common pattern:<\/strong> Hot coffee cooling to room temperature, pizza cooling after leaving the oven<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Step 1:<\/strong> Identify what&#8217;s changing (salt amount, temperature, etc.)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Step 2:<\/strong> Set up the rate equation based on physical principles<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Step 3:<\/strong> Apply separation of variables<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Step 4:<\/strong> Use given information to find constants<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Step 5:<\/strong> Answer the specific question asked<\/li>\n<\/ul>\n<p>Units matter! Make sure rates, concentrations, and times all work together.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1170571290129\" data-type=\"problem\">\n<p id=\"fs-id1170571290132\">A tank contains [latex]3[\/latex] kilograms of salt dissolved in [latex]75[\/latex] liters of water. A salt solution of [latex]0.4\\text{kg salt\/L}[\/latex] is pumped into the tank at a rate of [latex]6\\text{L\/min}[\/latex] and is drained at the same rate. Solve for the salt concentration at time [latex]t[\/latex]. Assume the tank is well mixed at all times.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558891\">Hint<\/button><\/p>\n<div id=\"q44558891\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573356102\" data-type=\"commentary\" data-element-type=\"hint\">\n<p>Follow the steps in the example:\u00a0Determining Salt Concentration over Time and determine an expression for INFLOW and OUTFLOW. Formulate an initial-value problem, and then solve it.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558892\">Show Solution<\/button><\/p>\n<div id=\"q44558892\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170571096071\" data-type=\"solution\">\n<p id=\"fs-id1170571096073\">Initial value problem:<\/p>\n<p id=\"fs-id1170571096076\">[latex]\\frac{du}{dt}=2.4-\\frac{2u}{25},u\\left(0\\right)=3[\/latex]<\/p>\n<p id=\"fs-id1170571145647\">[latex]\\text{Solution:}u\\left(t\\right)=30 - 27{e}^{\\frac{\\text{-}t}{50}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1170571110826\" data-type=\"problem\">\n<p id=\"fs-id1170571110828\">A cake is removed from the oven after baking thoroughly, and the temperature of the oven is [latex]450^\\circ\\text{F}\\text{.}[\/latex] The temperature of the kitchen is [latex]70^\\circ\\text{F}[\/latex], and after [latex]10[\/latex] minutes the temperature of the cake is [latex]430^\\circ\\text{F}\\text{.}[\/latex]<\/p>\n<ol id=\"fs-id1170571143291\" type=\"a\">\n<li>Write the appropriate initial-value problem to describe this situation.<\/li>\n<li>Solve the initial-value problem for [latex]T\\left(t\\right)[\/latex].<\/li>\n<li>How long will it take until the temperature of the cake is within [latex]5^\\circ\\text{F}[\/latex] of room temperature?<\/li>\n<\/ol>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558879\">Hint<\/button><\/p>\n<div id=\"q44558879\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170571137906\" data-type=\"commentary\" data-element-type=\"hint\">\n<p>Determine the values of [latex]{T}_{s}[\/latex] and [latex]{T}_{0}[\/latex] then use the example:\u00a0Solving an Initial-Value Problem.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558889\">Show Solution<\/button><\/p>\n<div id=\"q44558889\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170571292457\" data-type=\"solution\">\n<ol id=\"fs-id1170571292459\" type=\"a\">\n<li style=\"text-align: left;\">Initial-value problem<span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{dT}{dt}=k\\left(T - 70\\right),T\\left(0\\right)=450[\/latex]<\/li>\n<li>[latex]T\\left(t\\right)=70+380{e}^{kt}[\/latex]<\/li>\n<li>Approximately [latex]114[\/latex] minutes.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":21,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":669,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/826"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/826\/revisions"}],"predecessor-version":[{"id":2248,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/826\/revisions\/2248"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/669"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/826\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=826"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=826"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=826"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=826"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}