{"id":254036,"date":"2025-07-12T09:50:52","date_gmt":"2025-07-12T09:50:52","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=254036"},"modified":"2025-07-12T09:50:54","modified_gmt":"2025-07-12T09:50:54","slug":"for-0-to-24-hours-the-temperature-inside-a-refrigerator-in-a-kitchen-is-given-by-the-function-that-satisfies-the-differential-equation","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/12\/for-0-to-24-hours-the-temperature-inside-a-refrigerator-in-a-kitchen-is-given-by-the-function-that-satisfies-the-differential-equation\/","title":{"rendered":"\u00a0For\u00a00 to 24\u00a0hours, the temperature inside a refrigerator in a kitchen is given by the function\u00a0\u00a0that satisfies the differential equation\u00a0"},"content":{"rendered":"\n
 For\u00a00 to 24\u00a0hours, the temperature inside a refrigerator in a kitchen is given by the function\u00a0\u00a0that satisfies the differential equation\u00a0<\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/p>\n\n\n\n
Here are the correct answers and explanation for the problem.<\/p>\n\n\n\n
Correct Answer<\/h3>\n\n\n\n
(a)<\/strong> The equation of the tangent line is W = 0.5t + 3. Using this line, the approximate temperature at t = 0.4 hour is W(0.4) \u2248 3.2\u00b0C.<\/p>\n\n\n\n
(b)<\/strong> The particular solution is W(t) = \u221a(3 sin(t) + 9).<\/p>\n\n\n\n
(c)<\/strong> The expression for the total cost of operating the refrigerator for the 24 hour interval is:
Cost = \u222b[from 0 to 24] 0.001(20 – \u221a(3 sin(t) + 9)) dt<\/p>\n\n\n\n
\n\n\n\n
Explanation<\/h3>\n\n\n\n
This problem involves analyzing a differential equation that models the temperature inside a refrigerator. The solution requires applying calculus techniques for tangent line approximation, solving differential equations, and setting up a definite integral for a real world application.<\/p>\n\n\n\n
Part (a): Tangent Line Approximation<\/strong><\/p>\n\n\n\n
To find the equation of the tangent line at t = 0, we first need a point and a slope. The initial condition gives us the point (t, W) = (0, 3). The slope of the tangent line is the value of the derivative dW\/dt at this point. We substitute t = 0 and W = 3 into the given differential equation:<\/p>\n\n\n\n
dW\/dt = (3 cos(0)) \/ (2 * 3) = (3 * 1) \/ 6 = 1\/2.<\/p>\n\n\n\n
Using the point-slope form, W – W\u2081 = m(t – t\u2081), the tangent line equation is W – 3 = (1\/2)(t – 0), which simplifies to W = 0.5t + 3. To approximate the temperature at t = 0.4 hour, we substitute t = 0.4 into this linear equation: W(0.4) \u2248 0.5(0.4) + 3 = 0.2 + 3 = 3.2\u00b0C.<\/p>\n\n\n\n
Part (b): Particular Solution<\/strong><\/p>\n\n\n\n
To find the particular solution W(t), we use the method of separation of variables on the differential equation dW\/dt = (3 cos(t)) \/ (2W). We rearrange the equation to group W terms with dW and t terms with dt:<\/p>\n\n\n\n
2W dW = 3 cos(t) dt.<\/p>\n\n\n\n
Next, we integrate both sides:
\u222b 2W dW = \u222b 3 cos(t) dt
W\u00b2 = 3 sin(t) + C<\/p>\n\n\n\n
We find the constant of integration, C, using the initial condition W(0) = 3. Substituting these values gives 3\u00b2 = 3 sin(0) + C, which simplifies to 9 = 0 + C, so C = 9. The particular solution is W\u00b2 = 3 sin(t) + 9. Solving for W(t) yields W(t) = \u00b1\u221a(3 sin(t) + 9). Since the initial temperature W(0) = 3 is positive, we select the positive root, giving the final answer W(t) = \u221a(3 sin(t) + 9).<\/p>\n\n\n\n
Part (c): Cost Integral<\/strong><\/p>\n\n\n\n
The total cost is found by integrating the rate of cost over the 24 hour period. The cost accumulates at $0.001 per hour for each degree the kitchen temperature (20\u00b0C) exceeds the refrigerator’s temperature, W(t). The temperature difference at any time t is 20 – W(t). Therefore, the rate of cost in dollars per hour is 0.001 * (20 – W(t)). To find the total cost over the interval from t = 0 to t = 24, we integrate this rate:<\/p>\n\n\n\n
Cost = \u222b[from 0 to 24] 0.001(20 – W(t)) dt.<\/p>\n\n\n\n
Substituting the expression for W(t) from part (b), we get the final integral expression for the total cost.<\/p>\n\n\n\n
\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
For\u00a00 to 24\u00a0hours, the temperature inside a refrigerator in a kitchen is given by the function\u00a0\u00a0that satisfies the differential equation\u00a0 The Correct Answer and Explanation is: Here are the correct answers and explanation for the problem. Correct Answer (a) The equation of the tangent line is W = 0.5t + 3. Using this line, the approximate temperature […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-254036","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/254036","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=254036"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/254036\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=254036"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=254036"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=254036"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}