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 76 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)=76+226e^(-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

Part 1:

98.6 - 2 = 76 + 226 * e^(-k * 1)

96.6 - 76 = 226 * e^(-k)
20.6 / 226 = e^(-k)
ln(20.6) - ln(226) = -k
ln(226) - ln(20.6) = k

2.39 = k


Part 2:

T = 76 + 226 * e^(ln(20.6 / 226) * t)
dT/dm = 226 * ln(20.6 / 226) * e^(ln(20.6 / 226) * t) * dt/dm

dT/dm = -1
dt/dm = 1

-1 = 226 * ln(20.6 / 226) * 1 * e^(ln(20.6 / 226) * t)
-1 / (226 * ln(20.6 / 226)) = e^(ln(20.6 / 226) * t)
ln(-1 / (226 * ln(20.6) - 226 * ln(226)) = ln(20.6 / 226) * t
ln(1 / (32.6 * ln(226/20.6)) = ln(20.6 / 226) * t
-ln(32.6 * ln(226/20.6)) / ln(20.6 / 226) = t
1.819 = t

So, after about 1 hour and 48 minutes 36 secnds, the body will be cooling at about 1 degree per hour

Part 3:


The coroner is expecting the body to cool 25 degrees in 24 hours. If we plug in our value for -k (ln(306/326)) and t = 24, then we should get a close agreement

Rule of thumb: 98.6 - 25 = 73.6

T = 76 + 226 * e^(ln(20.6 / 226) * 24)
T = 76

So, the coroner would have been off by only 2.4 degrees just using his rule of thumb calculation.

Hope this helped. Please rate 5 if it did.
Cheers, mate!

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