Explanation Given: The heated bar is cooled until the temperature is 100 ° F , while the water is at a constant temperature 50 ° F . Theorem used: Separation of Variables: The solution of the differential equation, f ( y ) y ′ = g ( x ) is given implicitly by the equation, ∫ f ( y ) d y = ∫ g ( x ) d x . Formula used: Limited growth model: If the rate of change with respect to time t of a quantity y is proportional to the difference between y and a limiting value M , then y satisfies the differential equation, d y d t = k ( M − y ) , k > 0 , t > 0 Calculation: Let y represents the temperature of the bar at time t . Given that, the water is maintained at a constant temperature of 50 ° F . Thus, the rate is proportional to the difference of the temperature and the limit 50 ° F , which is d y d t = k ( y − 50 ) , k > 0 , t > 0 . Thus, the problem follows a limited growth model. Rewrite the differential equation d y d t = k ( y − 50 ) as follows. d y d t = k ( y − 50 ) 1 y − 50 d y = k d t The given differential equation d y d t = k ( y − 50 ) is in the form of f ( y ) y ′ = g ( x ) where, f ( y ) = 1 ( y − 50 ) and g ( t ) = k . By the above theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ 1 ( y − 50 ) d y = ∫ k d t ln ( y − 50 ) = k t + C 1 e ln ( y − 50 ) = e k t + C 1 Simplify further as, y − 50 = e k t + C 1 y = 50 + e k t + C 1 y = 50 + C e k t , where C = e C 1 The bar is at 500 ° F and cooled to 400 ° F after 5 minutes

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 9.2, Problem 72E

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To find: The time for the bar to be left in water.

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Chapter 9.2, Problem 71E

Chapter 9.2, Problem 73E

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Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

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The time for the bar to be left in water.

Solution Summary: Theorem used: Separation of Variables: If the rate of change with respect to time t of a quantity y is proportional to the difference of the temperature and the limit

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