Newton\'s law of cooling states that the temperature of an object changes at a r

Question

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. The coffee has a temperature of 185 degrees Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 172 degrees in a room at 64 degrees.

b) Determine when the coffee reaches a temperature of 127 degrees.

The coffee will reach a temperature of 127 degrees in------------------ minutes

Explanation / Answer

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) = 64+(185-64)*e^(-kt)
T(t) = 64+121^(-kt)
172=64+121*e^-2.5k
108=121*e^(-2.5k)
ln(108/121) = -2.5k
k = 0.0454
T(t) = 64 + 121*e^(-0.0454t)
127 = 64 + 121*e^-0.0454t)
63/121 = e^(-0.0454t)
ln(63/121) = -0.0454t
t = 14.37 minutes

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