Chapter 11.1, Problem 58E

Newton’s Law of Cooling Newton’s law of cooling states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and the surrounding medium. Thus, if T is the temperature of the object after t hours and $T_M$ is the (constant) temperature of the surrounding medium, then

$\frac{dT}{dt}=-k\left(T-T_M\right)$

where k is a constant. Use this equation in Exercises 58-61.

Show that the solution of this differential equation is

$T=C{e}^{-kt}+T_M$

where C is a constant.