{"id":3431,"date":"2024-06-24T16:39:43","date_gmt":"2024-06-24T16:39:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3431"},"modified":"2025-08-17T23:25:21","modified_gmt":"2025-08-17T23:25:21","slug":"related-rates-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/related-rates-apply-it\/","title":{"raw":"Related Rates: Apply It","rendered":"Related Rates: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Show how quantities change using derivatives and explore how these changes are connected<\/li>\r\n\t<li>Apply the chain rule to calculate how one changing quantity affects another<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Coffee Cooling Rate Analysis: Exploring Related Rates<\/h2>\r\n<p class=\"whitespace-pre-wrap break-words\">In this apply-it task, we'll investigate the cooling rate of coffee in various containers, applying concepts of derivatives and the chain rule to real-world scenarios. This will help us understand how different factors affect the rate of temperature change over time.<\/p>\r\n<center>\r\n[caption id=\"attachment_3980\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-3980\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16175307\/coffee-869203_1280.jpg\" alt=\"Image of coffee cup\" width=\"500\" height=\"333\" \/> Cup of coffee[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Given: The temperature of coffee [latex]T(t)[\/latex] over time [latex]t[\/latex] is given by: [latex]T(t) = T_{a} + (T_{0} - T_{a})e^(-kt)[\/latex], where:<\/p>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]T(t)[\/latex] is the temperature of coffee at time t min,<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]T_{a}[\/latex] is the ambient temperature,<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]T_{0}[\/latex] is the initial temperature of the coffee,<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]k[\/latex] is the cooling constant specific to the container,<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]t[\/latex] is time in minutes.<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]288233[\/ohm_question]<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]288234[\/ohm_question]<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]288235[\/ohm_question]<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]288236[\/ohm_question]<\/section>\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]288237[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Show how quantities change using derivatives and explore how these changes are connected<\/li>\n<li>Apply the chain rule to calculate how one changing quantity affects another<\/li>\n<\/ul>\n<\/section>\n<h2>Coffee Cooling Rate Analysis: Exploring Related Rates<\/h2>\n<p class=\"whitespace-pre-wrap break-words\">In this apply-it task, we&#8217;ll investigate the cooling rate of coffee in various containers, applying concepts of derivatives and the chain rule to real-world scenarios. This will help us understand how different factors affect the rate of temperature change over time.<\/p>\n<div style=\"text-align: center;\">\n<figure id=\"attachment_3980\" aria-describedby=\"caption-attachment-3980\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3980\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16175307\/coffee-869203_1280.jpg\" alt=\"Image of coffee cup\" width=\"500\" height=\"333\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16175307\/coffee-869203_1280.jpg 1280w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16175307\/coffee-869203_1280-300x200.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16175307\/coffee-869203_1280-1024x682.jpg 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16175307\/coffee-869203_1280-768x512.jpg 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16175307\/coffee-869203_1280-1200x800.jpg 1200w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16175307\/coffee-869203_1280-65x43.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16175307\/coffee-869203_1280-225x150.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/06\/16175307\/coffee-869203_1280-350x233.jpg 350w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-3980\" class=\"wp-caption-text\">Cup of coffee<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Given: The temperature of coffee [latex]T(t)[\/latex] over time [latex]t[\/latex] is given by: [latex]T(t) = T_{a} + (T_{0} - T_{a})e^(-kt)[\/latex], where:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]T(t)[\/latex] is the temperature of coffee at time t min,<\/li>\n<li class=\"whitespace-normal break-words\">[latex]T_{a}[\/latex] is the ambient temperature,<\/li>\n<li class=\"whitespace-normal break-words\">[latex]T_{0}[\/latex] is the initial temperature of the coffee,<\/li>\n<li class=\"whitespace-normal break-words\">[latex]k[\/latex] is the cooling constant specific to the container,<\/li>\n<li class=\"whitespace-normal break-words\">[latex]t[\/latex] is time in minutes.<\/li>\n<\/ul>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm288233\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288233&theme=lumen&iframe_resize_id=ohm288233&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm288234\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288234&theme=lumen&iframe_resize_id=ohm288234&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm288235\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288235&theme=lumen&iframe_resize_id=ohm288235&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm288236\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288236&theme=lumen&iframe_resize_id=ohm288236&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm288237\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288237&theme=lumen&iframe_resize_id=ohm288237&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":652,"module-header":"apply_it","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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3431"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":11,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3431\/revisions"}],"predecessor-version":[{"id":4775,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3431\/revisions\/4775"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/652"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3431\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3431"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3431"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3431"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3431"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}