Coroners estimate time of death using the rule of thumb that a body cools about 2^◦F during the first hour after death and about 1^◦F for each additional hour. Assuming an air temperature of 68^◦F and a living body temperature of 98.6^◦F, the temperature T (t) in ^◦F of a body at a time t hours since death is given by

T (t) = 68 + 30.6e^−kt.

(a) For what value of k will the body cool by 2^◦F in the first hour?
(b) Using the value of k found in part (a), after how many hours will the temperature of the body be decreasing at a rate of 1^◦F per hour?
(c) Using the value of k found in part (a), show that, 24 hours after death, the coroner’s rule of thumb gives approximately the same temperature as the formula.