Chapter 28, Problem 9P

In the investigation of a homicide or accidental death, it isoften important to estimate the time of death. From the experimental observations, it is known that the surface temperature of anobject changes at a rate proportional to the difference between the temperature of the object and that of the surrounding environmentor ambient temperature. This is known as Newton's law of cooling. Thus, if $T\left(t\right)$ is the temperature of the object at time t, and $T_a$ is the constant ambient temperature:

$\frac{dT}{dt}=-K\left(T-T_a\right)$

where $K>0$ is a constant of proportionality. Suppose that at time $t=0$ a corpse is discovered and its temperature is measured to be $T_0$. We assume that at the time of death, the body temperature, $T_d$, was at the normal value of $37°\text{C}$. Suppose that the temperature of the corpse when it was discovered was $29.5°\text{C}$, and that two hourslater, it is $23.5°\text{C}$. The ambient temperature is $20°\text{C}$.

(a) Determine K and the time of death.

(b) Solve the ODE numerically and plot the results.