The room temperature in your office is 70 degree Fahrenheit. Experience has taug

Question

The room temperature in your office is 70 degree Fahrenheit. Experience has taught you that the temperature of a cup of coffee brought to your office will drop from 120 degree Fahrenheit to 100 degree Fahrenheit in 10 minutes. What should be the temperature of your cup of coffee if you want it to take 20 minutes before it drops to 100 degree Fahrenheit?

(a) 120 degree Fahrenheit approx
(b) 153 degree Fahrenheit approx
(c) 100 degree Fahrenheit approx
(d) 60 degree Fahrenheit approx
(e) None of these

Explanation / Answer

Q1)  We use Newton's Law of Cooling:

dT/dt = -k(T-Ta)

Where T is the temperature, Ta is the ambient temperature (70), and k is the constant of proportionality.  Let's solve this.

dT/dt = -k(T-70) 

say u = T-70, then du/dt = dT/dt

du/dt = -ku

1/u du = -k dt

ln(u) = -kt + C

u = e-kt+C

u = Ae-kt

T - 70 = Ae-kt

T = 70 + Ae-kt

Now, we need to find A and k.  Let's use our boundary condition. 

T(0) = 70 + A = 120

A = 50

T(10) = 70 + 50e-k(10) = 100

50e-10k = 30

e-10k = 3/5

-10k = ln(3/5)

k = -ln(3/5)/10 = 0.0511

So, T(t) = 70 + 50e-0.0511t

So, we know T(10) = 100.  If it takes 20 minutes to get there, let's say that t = -10

T(-10) = 70 + 50e-0.0511*-10 = 153.3°F

So, the answer is B

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