Explanation Given: The room temperature is 75 ° F and hot coal is pulled out of a furnace and allowed to cool in the room temperature, after 10 min , the temperature of the coal was 415 ° F and after 20 min the temperature was 347 ° F . Approach: The Newton’s law of cooling is given below, d T d t = − k ( T − T m ) Here, T m is room temperature, d T d t is rate of change of temperature and T is coal temperature. Calculation: Consider the expression for Newton’s law of cooling. d T d t = − k ( T − T m ) Substitute 75 ° F for T m in the above equation and simply the equation. d T ( T − 75 ) = − k d t Integrate the above equation. ∫ d T ( T − 75 ) = ∫ − k d t ln ( T − 75 ) = − k t + c ... ( 1 ) Take exponential both sides. Substitute 10 min for t and 415 ° F for T in the equation ( 1 ) to calculate the value of constant. 415 = 75 + c e − 10 k c = 347 e − 10 k ... ( 2 ) Substitute 20 min for t and 347 ° F for T in the equation ( 1 ) to calculate the value of constant To determine (b) The time when the temperature of the coal is 100 ° F .

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Chapter 1.4, Problem 27P

To determine

(a)

The temperature of the furnace.

To determine

(b)

The time when the temperature of the coal is 100°F.

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Chapter 1.4, Problem 26P

Chapter 1.4, Problem 28P

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Chapter 1 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 1.1 - True-False review For items a-n, decide if the...Ch. 1.1 - For Problems 14 determine the order of the...Ch. 1.1 - For Problems 14 determine the order of the...Ch. 1.1 - For Problems 1-4 determine the order of the...Ch. 1.1 - For Problems 14 determine the order of the...Ch. 1.1 - Verify that, for t0, y(t)=lnt is a solution to the...Ch. 1.1 - Verify that y(x)=x/(x+1) is a solution to the...Ch. 1.1 - Verify that y(x)=exsinx is a solution to the...Ch. 1.1 - By writing Equation 1.1.7 in the form 1TTmdTdt=k...Ch. 1.1 - A glass of water whose temperature is 50F is taken...

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(a) The temperature of the furnace.

Solution Summary: The author explains how to determine the temperature of the furnace.