In the investigation of a homicide or accidental death, it is often important to estimate...

In the investigation of a homicide or accidental death, it is often important to estimate the time of death. From the experimental observations, it is known that the surface temperature of an object changes at a rate proportional to the difference between the temperature of the object and that of the surrounding environment or ambient temperature. This is known as Newton’s law of cooling. Thus, if T(t) is the temperature of the object at time t, and Ta is the constant ambient temperature:

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 To. We assume that at the time of death, the body temperature Td was at the normal value of 37 °C.

Suppose that the temperature of the corpse when it was discovered was 29.5 °C, and that two hours later, it is 23.5 °C. The ambient temperature is 20 °C.

(a) Determine K and the time of death.

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