Coroners estimate time of death using the rule of thumb that a body cools about
ID: 2892805 • Letter: C
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 78 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) = 78 + 20.6 e^{-kt}.
1. For what value of k will the body cool by 2 degrees F in the first hour?
k=
2. 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
3. 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
Coroners estimate time of death:
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
-... -2.0601 e^(-0.05954t)
1 / 2.0601 e^(-0.05954t)
ln(1 / 2.0601) -0.05954t
ln(1 / 2.0601) / -0.05954 t
... 12.1390 hrs
3) Coroner's estimate:
after 24 hours, body has cooled by 2 + 23 degrees = 25°
so 98.6 - 25 = 73.6°
and per T(t):
T(24) 64 + 34.6 e^(-0.05954(24))
T(24) 64 + 34.6 e^(-1.42899)
T(24) 72.3°
So
coroner would estimate ToD 1h 18m earlier than T(t)