Newton\'s law of cooling states that the temperature of an object changes at a r
ID: 2880537 • 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 190 degrees Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 178 degrees in a room at 68 degrees, determine when the coffee reaches a temperature of 143degrees. The coffee will reach a temperature of 143 degrees in minutes.Explanation / Answer
Let T(t) is the temperature of coffee at any time t,
then, 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+(190-68)*e^(-kt) [given T(0)=To=190, S=68, T(t)=143, T(2.5)=178]
=>T(t) = 68+132^(-kt)
=>178=68+132*e^-2.5k
=>110=132*e^(-2.5k)
=> ln(110/132) = -2.5k
=>k = 0.0729
Now, T(t) = 68 + 132*e^(-0.0729t)
=>143 = 68 + 32*e^-0.0729t)
=>77/132 = e^(-0.0729t)
=>ln(77/132) = -0.0729t
t = 7.39 minutes