Explanation Given: Initial temperature of the thermometer is 70 ° F and the air temperature outside is 10 ° F . Calculation: Consider the equation for the Newton’s Law of cooling. d T d t = k ( T − 10 ) (1) Here, k is a constant. Initial temperature in the thermometer is 70 ° F . Thus, T ( 0 ) = 70 . After one-half minute the thermometer reads 50 ° F . Thus, T ( 1 2 ) = 50 . Separate the variables in equation (1) and integrate. ∫ 1 T − 10 d T = ∫ k d t ln | T − 10 | = k t + c 0 | T − 10 | = e k t + c 0 T − 10 = ± e c 0 e k t Simplify the above equation. T − 10 = C e k t T ( t ) = 10 + C e k t (2) Substitute 0 for t and T ( 0 ) = 70 in equation (2). T ( 0 ) = 10 + C e k ( 0 ) 70 = 10 + C e 0 70 = 10 + C ⋅ 1 C = 70 − 10 Simplify the above equation. C = 60 Substitute 60 for C in equation (2). T ( t ) = 10 + 60 e k t (3) Substitute 1 2 for t and T ( 1 2 ) = 50 in equation (3). T ( 1 2 ) = 10 + 60 e k ( 1 2 ) 50 = 10 + 60 e 1 2 k 60 e 1 2 k = 40 e 1 2 k = 4 6 Take natural logarithm on both sides of the above equation

6th Edition

ISBN: 9781284127003

Author: ZILL

Publisher: JONES+BARTLETT LEARNING,LLC (CC)

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Chapter 2.7, Problem 13E

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Determine the reading of the thermometer at t=1 min and the time taken by thermometer to reach 15° F.

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Chapter 2.7, Problem 12E

Chapter 2.7, Problem 14E

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Determine the reading of the thermometer at t = 1 min and the time taken by thermometer to reach 15 ° F .

Solution Summary: The author calculates the reading of the thermometer at t=1mathrmmin, and the time taken by it to reach 15°.

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Determine the reading of the thermometer at t=1 min and the time taken by thermometer to reach 15° F.

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Chapter 2 Solutions