{"id":825,"date":"2025-06-20T17:15:59","date_gmt":"2025-06-20T17:15:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=825"},"modified":"2025-12-17T16:35:14","modified_gmt":"2025-12-17T16:35:14","slug":"separation-of-variables-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/separation-of-variables-apply-it\/","title":{"raw":"Separation of Variables: Apply It","rendered":"Separation of Variables: Apply It"},"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 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Community Garden Water Management<\/h2>\r\n<p class=\"whitespace-normal break-words\">You volunteer at a local community garden that serves families from diverse backgrounds. The garden has a 200-gallon water storage tank that collects rainwater and receives treated water from the city. Your job is to monitor the water quality by tracking the concentration of nutrients that are added to help the plants grow.<\/p>\r\n<p class=\"whitespace-normal break-words\">Initially, the tank contains 200 gallons of water with [latex]3[\/latex] pounds of dissolved nutrients. At [latex]t = 0[\/latex] (measured in hours), the garden begins receiving city water that flows into the tank at a rate of [latex]5[\/latex] gallons per hour. This incoming water contains nutrients at a concentration of [latex]0.2[\/latex] pounds per gallon. To maintain proper circulation, water flows out of the tank at the same rate of [latex]5[\/latex] gallons per hour, keeping the volume constant at 200 gallons. The water in the tank is continuously mixed by a solar-powered pump.<\/p>\r\n<p class=\"whitespace-normal break-words\">Later in the day, you also need to monitor the temperature of water in a small greenhouse pool used for starting seedlings. The pool water needs to cool from its daytime temperature to be suitable for sensitive plants overnight.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313469[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313470[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313472[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">[ohm_question hide_question_numbers=1]313474[\/ohm_question]<\/p>\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 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Community Garden Water Management<\/h2>\n<p class=\"whitespace-normal break-words\">You volunteer at a local community garden that serves families from diverse backgrounds. The garden has a 200-gallon water storage tank that collects rainwater and receives treated water from the city. Your job is to monitor the water quality by tracking the concentration of nutrients that are added to help the plants grow.<\/p>\n<p class=\"whitespace-normal break-words\">Initially, the tank contains 200 gallons of water with [latex]3[\/latex] pounds of dissolved nutrients. At [latex]t = 0[\/latex] (measured in hours), the garden begins receiving city water that flows into the tank at a rate of [latex]5[\/latex] gallons per hour. This incoming water contains nutrients at a concentration of [latex]0.2[\/latex] pounds per gallon. To maintain proper circulation, water flows out of the tank at the same rate of [latex]5[\/latex] gallons per hour, keeping the volume constant at 200 gallons. The water in the tank is continuously mixed by a solar-powered pump.<\/p>\n<p class=\"whitespace-normal break-words\">Later in the day, you also need to monitor the temperature of water in a small greenhouse pool used for starting seedlings. 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