2) Newton\'s law of cooling states that the temperature of an object changes at

Question

2) Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 200 degrees Fahrenheit when freshly poured, and 1.5minutes later has cooled to 182 degrees in a room at 68 degrees, determine when the coffee reaches a temperature of 137 degrees.
The coffee will reach a temperature of 137 degrees in           miutes

Explanation / Answer

The temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings

dT/dt = -k(T-S) where T is current temperature and S = ambient temperature
dT/(T-S) = -k.dt
Solving the differential equation gives
ln(T-S) = -kt + C
T-S = e^(-kt+C)
T(t) = S + e^(-kt+C)
T(t) = S +(To-S)*e(-kt) where To = initial temperature at t = 0
T(t) = 68+(200-68)*e^(-kt)
T(t) = 68+132^(-kt)
182=68+132*e^-1.5k
114=132*e^(-1.5k)
ln(114/132) = -1.5k
k = 0.0977

T(t) = 68 + 132*e^(-0.0977t)
137 = 68 + 132*e^-0.0977t)
69/132 = e^(-0.0977t)
ln(68/132) = -0.0977t
t = 6.79 minutes

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