Explanation Given information: The rate of cooling of the object by Newton’s law of cooling is given as, d u d t = k ( u − T ) where T is the temperature of the surroundings. The room temperature is 20 ° C and initial temperature of the object is 0 ° C . Formula used: The Newton’s law of cooling is given as, d u d t = k ( u − T ) Where T is the temperature of the surroundings u is the initial temperature, t is any instantaneous time. The logarithmic rule of integrals, ∫ 1 x d x = ln | x | + C where x ≠ 0 . The natural logarithm property, log a b = y ⇒ b = a y . Calculation: Consider the rate of cooling of the object, d u d t = k ( u − T ) . Rewrite the provided differential equation, d u u − T = k d t Integrate both side of the equation as: ∫ d u u − T = ∫ k d t Now, use the logarithmic rule of integrals to get, ∫ d u u − T = ∫ k d t ln | u − T | = k t + C u − T = e k t + C Solve further, u − T = e k t ⋅ e C u − T = A e k t u = T + A e k t Now, initial temperature of the object is 0 ° C and initial temperature of the room is 20 ° C

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Differential Equations

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Chapter 12.5, Problem 57E

To determine

To calculate: The time require by the object to reach the temperature of 18° C for which the rate of cooling of the object by Newton’s law of cooling is given as, dudt=k(u−T) where T is the temperature of the surroundings.

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Chapter 12 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 12.1 - 1. True or false: (a) (b) Ch. 12.1 - Prob. 2CP Ch. 12.1 - 1. If what is Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E Ch. 12.1 - Prob. 5E Ch. 12.1 - Evaluate the integrals in Problems 5-28. Check...Ch. 12.1 - Prob. 7E Ch. 12.1 - Evaluate the integrals in Problems 5-28. Check...

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The time require by the object to reach the temperature of 18 ° C for which the rate of cooling of the object by Newton’s law of cooling is given as, d u d t = k ( u − T ) where T is the temperature of the surroundings.

Solution Summary: The author calculates the time required by the object to reach the temperature of 18°C based on Newton's law of cooling.

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To calculate: The time require by the object to reach the temperature of 18° C for which the rate of cooling of the object by Newton’s law of cooling is given as, dudt=k(u−T) where T is the temperature of the surroundings.

Chapter 12 Solutions