Chapter 10.4, Problem 15E

Determining the Time of Death A body was found in a room when the room's temperature was ${70}^{0}F$. Let $f(t)$ denote the temperature of the body $t$ hours from the time of death. According to Newton's law of cooling, $f$ satisfies a differential equation of the form

$y\text{'}=k\left(T-y\right)$.

a. Find T.

b. After several measurements of the body's temperature, it was determined that when the temperature of the body was $80$ degrees, it was decreasing at the rate of $5$ degrees per hour. Find $k$.

c. Suppose that, at the time of death, the body's temperature was about normal, say, ${98}^0\text{F}$. Determine $f(t)$.

d. When the body was discovered, its temperature was ${85}^0\text{F}$. Determine how long ago the person died.