Newton\'s law of cooling states that the temperature of an object changes at a r
ID: 2965454 • Letter: N
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. If the coffee has a temperature of 185 degrees Fahrenheit when freshly poured, and 2 minutes later has cooled to 175 degrees in a room at 66 degrees, determine when the coffee reaches a temperature of 150 degrees. The coffee will reach a temperature of 150 degrees in _______ minutes
Explanation / Answer
We know that,
T/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) = 66+(185-66)*e^(-kt)
T(t) = 66+119*e^(-kt)
175=66+119*e^-2k
109=119*e^(-2k)
ln(109/119) = -2k
k = 0.04389
Now,
T(t) = 66 + 119*e^(-0.04389t)
150 = 66 + 119*e^(-0.04389t)
84/119 = e^(-0.04389t)
ln(84/119) = -0.04389t
t = 7.9359 minutes