Newton\'s law of cooling states that the temperature of an object changes at a r
ID: 2981495 • 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 195 degrees Fahrenheit when freshly poured, and 2 minutes later has cooled to 182 degrees in a room at 80 degrees, determine when the coffee reaches a temperature of 152 degrees. The coffee will reach a temperature of 152 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
182 = 80 + (195-80) e^(-2k)
k = 0.0455 min^-1
now
152 = 80 + (195-80)e^(-kt)
e^-kt = 152-80 / 195-80
-kt = ln(72/115)
t = 0.4683 / 0.0455 = 10.29 = 10.3 min