Newton\'s law of cooling states that the temperature of an object changes at a r
ID: 1890081 • 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 210 degrees Fahrenheit when freshly poured, and 3 minutes later has cooled to 191 degrees in a room at 76 degrees, determine when the coffee reaches a temperature of 161 degrees.The coffee will reach a temperature of 161 degrees in what 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) = 76+(210-76)*e^(-kt) T(t) = 76+134^(-kt) 191=76+134*e^-3k 115=134*e^(-3k) ln(115/134) = -3k k = 0.050969224 T(t) = 76 + 134*e^(-0.050969224t) 161 = 76 + 134*e^-0.050969224t) 85/134 = e^(-0.050969224t) ln(85/134) = -0.050969224t t = 8.930654795 minutes