D5-Direction Fields: Problem 13 Previous Problem Problem List Next Problem (3 po

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D5-Direction Fields: Problem 13 Previous Problem Problem List Next Problem (3 points) Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Suppose t is time, T is the temperature of the object, and Ts is the surrounding temperature. The following differential equation describes Newton's Law dT dt where k is a constant Suppose that we consider a 95°C cup of coffee in a 24°C room. Suppose it is known that the coffee cools at a rate of 1°C/min. when it is 70°C. Answer the following questions. 1. Find the constant k in the differential equation. Answer (in per minute): k- 2. What is the limiting value of the temperature? Answer (in Celsius): T 3. Use Euler's method with step size h = 2 minutes to estimate the temperature of the coffee after 10 minutes. Answer (in Celsius): T(10) Note: You can earn partial credit on this problem.

Explanation / Answer

dT/dt = k(T - Ts)

Now, we are given that dT/dt = -1 when T = 70 :
So,
-1 = k(70 - 24)
-1 = k*46
k = -1/46 ---> ANS 1

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dT/dt = k(T - 24)

dT/(T - 24) = kdt

Integrating :
ln(T - 24) = kt + C

T - 24 = e^(kt + C)

T = 24 + Ce^(kt)

Now using T(0) = 95, we have
95 = 24 + Ce^0
95 - 24 = C
C = 71

So,
T = 24 + 71e^(kt)

Plug in k = -1/46 :
T = 24 + 71e^(-t/46)

Limiting value :
As t ---> inf, we get e^(-t/46) = 0

So, T = 24 + 0

T = 24 ----> ANS 2

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3)
Plug in t = 10 :

T = 24 + 71e^(-t/46)

T = 24 + 71e^(-10/46)

81.128 ----> ANS 3

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