Question

It is a dark clear night. The air temperature is 15◦ C. A body is discovered at
midnight. Its temperature is 23◦ C. One hour later, the body has cooled to 20◦ C. Use Newton’s
law of cooling to determine the time of death.

Answer 1

The temperature T is the ambient temperature plus a steadily decreasing difference. In this case,

T(t) = 15 + (23-15)e^(-kt)
You can see that
T(0) = 15 + (23-15)*1 = 23
that is the starting temperature. Now we are told that T(1) = 20:

15+8e^-k = 20
k = 0.47

So, T(t) = 15+8e^(-0.47t)

When did death occur? When the body had normal temperature: 37°C

15+8e^(-0.47t) = 37
t = -2.1523

So, 2.1523 hours before midnight, death arrived. That is, at

9:50:52 pm