coffee is best enjoyed at Choose the one alternative that best completes the sta

Question

coffee is best enjoyed at Choose the one alternative that best completes the statement or answers the question Coffee is best enjoyed at a temperature of 125 degree F. A restaurant owner wants to discover the temperature T at which he should serve his coffee so that it will have cooled to this ideal temperature in 5 minutes. He discovers that a cup of coffee served at 196 degree F cools to 182 degree F in one minute when his restaurant is at 68 degree F. If he maintains the restaurant temperature at 68 degree F, at what temperature should he serve the coffee to meet his goal? A) 190 degree F B) 165 degree F C) 196 degree F D)170 degree F

Explanation / Answer

We can use Newton's law of cooling here...

T = Ta + (Th - Ta)e^(-kt)

where
T = Temperature at time t,
TA = Ambient temperature (temp of surroundings),
TH = Temperature of hot object at time 0,
k = positive constant and
t = time.

T = 68 + (Th - 68)e^(-kt)

When t = 0, Th = 196 and T = 182

182= 68 + (196 - 68)e^(-k*1)
114 = 128e^(-k)
0.890625 = e^-k

k = -ln(0.890625)

k = 0.1158318155251217

So, equation is :
T = 68 + (Th - 68)e^(-0.1158t)

125 = 68 + (Th - 68)e^(-0.1158*5)

57 = (Th - 68) * (0.5604585449744904)

57 / 0.5604585449744904 = Th - 68

101.7024372473335891296 + 68 = Th

Th = 169.72

Option D

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