Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 2 / 5A / b / i / Solution

With a positive cooling rate $k$, Newton's law of cooling gives $T^{\prime }=-k(T-T_0)$. The temperature excess $U=T-T_0$ satisfies the first-order linear differential equation $U^{\prime }=-kU$. Applying the initial condition to its exponential decay solution givesThe temperature approaches the ambient value $T_0$, rather than zero. The intended cooling situation has $T_0>0$ and $k>0$.