When a dead body is discovered, one of the first steps in the ensuing investigation is for a medical examiner to determine the time of death as closely as possible. Have you ever wondered how this is done? If the temperature of the medium (air, water, or whatever) has been fairly constant and less than 48 hours have passed since the death, Newton’s law of cooling can be used. The medical examiner does not actually solve the equation for each case. Instead, a table based on the formula is consulted. Use Newton’s law of cooling to work the following exercises.

Suppose the air temperature surrounding a body remains at a constant 10°F, C = 88.6, and k = 0.24, where time is measured in hours.
(a) Determine a formula for the temperature at any time t.
(b) Use a graphing calculator to graph the temperature T as a function of time t on the window [0, 30] by [0, 100].
(c) When does the temperature of the body decrease more rapidly: just after death, or much later? How do you know?

(d) What will the temperature of the body be after 4 hours?
(e) How long will it take for the body temperature to reach 40°F? Use your calculator graph to verify your answer.