Coroners estimate time of death using the rule of thumb that a body cools about

Question

Coroners estimate time of death using the rule of thumb that a body cools about 2 degrees F during the first hour after death and about 1 degree F for each additional hour. Assuming an air temperature of 64 degrees F and a living body temperature of 98.6 degrees F, the temperature T(t) in degrees F of a body at a time t hours since death is given by T(t) = 64 + 34.6 e^(-kt). a) For what value of k will the body cool by 2 degrees F in the first hour? k=? b) Using the value of k found above, after how many hours will the temperature of the body be decreasing at a rate of 1 degree F per hour? after ? hours c) Using the value of k found above, show by calculating both values that, 24 hours after death, the coroner's rule of thumb gives approximately the same temperature as the formula. T(24)= ? degrees F Rule of thumb gives: T = ? degrees F

Explanation / Answer

Given
T(t) = 64 + 34.6 e^(-kt)

1) given:
(t, T(t)) = (1, 96.6)
so
(96.6) = 64 + 34.6 e^(-k(1))
32.6/34.6 = e^(-k)
ln(32.6/34.6) = ln(e^(-k))
ln(32.6/34.6) = -k ln(e)
-ln(32.6/34.6) = k
0.05954 ? k

2) T(t) ? 64 + 34.6 e^(-0.05954t)
T'(t) ? -2.0601 e^(-0.05954t)
so
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