A thermometer is removed from a room where the air temperatu

A thermometer is removed from a room where the air temperature is 60 F to the outside where the temperature is 10 F. After 1 minute the thermometer reads 50 F . What is the reading at t=2 minute? how long will it take for the temperature to reach 20 F ?

Solution

Use Newton\'s law of cooling:
dT/dt = k(T - Ta) .......... Ta is the ambient temperature
dT/(T - Ta) = k dt
ln(T - Ta) = kt + c
T = Ce^(kt) + Ta

T[0] = 60, Ta = 10 .... C = 50
T[1] = 50 ... k = ln(4/9)

T = 60 (4/9)^t + 10
20= 50 (4/9)^t + 10
t 3.1 min