ewton\'s law of cooling states that the temperature of an object changes at a ra
ID: 2966358 • Letter: E
Question
ewton'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 190 degrees Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 174 degrees in a room at 78 degrees, determine when the coffee reaches a temperature of 139 degrees. The coffee will reach a temperature of 139 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) = 78+(190-78)*e^(-kt)
T(t) = 78+112^(-kt)
174=78+112*e^-2.5k
96=112*e^(-2.5k)
ln(96/112) = -2.5k
k = 0.0616
T(t) = 78 + 112*e^(-0.0616t)
139 = 78 + 112*e^-0.0616t)
61/112 = e^(-0.0616t)
ln(61/112) = -0.0616t
t = 9.86 minutes
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