Coroner's rule of thumb

[Guest]

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 additiona hour. Assuming an air tempeature of 68 degrees F and a living body temperature of 98.6 degree F, the temperature T(t) in degrees F of a body at a time t hour since death is given by
T(t)=68+30.6e^-kt.
a) for what value of k found in part a, will the body cool by 2 degrees 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 degree F per hour?
c)for what 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.

 

[stapel]

You are given:

. . . . .T(t) = 68 + 30.6e^-kt

...for time "t" in hours and temperature "T" in degrees Fahrenheit.

a) You are asked for the value of k, given that T(1) = T(0) - 2. Since T(0) is assumed to be 98.6, plug "98.6" in for "T(0)". Plug "1" in for "t" to find the formula for T(1). Subtract 2 from 98.6, and set this equal to your formula for T(1). Solve (using logs) for the value of the decay constant "k".

b) The rate of change of T(t) is dT/dt. So differentiate with respect to t, and set the result equal to "-1". Solve for the time t.

c) If you start at a temperature of 98.6°F, what will the temperature be, according to the rule of thumb, after one hour? What will the temperature be, according to the rule of thumb, after another twenty-three hours? What is the value, from the formula, of T(24)? Compare the results.

Hope this helps a bit. If you get stuck, please reply showing all the steps you have attempted. Thank you.

Eliz.

 

[Gene]

a) 96.6=68+30.6e^-k
28.6/30.6=e^-k
ln(28.6/30.6)=-k
k=.068

b) T=68+30.6e^(-.068t)
dT/dt = -30.6*.068*e^(-.068t) = -1
e^(-.068t)=1/(30.6*.068)
ln(e^(-.068t))=ln(1/(30.6*.068))
(-.068t)=-.733
t=10.8

c) The question makes no sense. They must be looking for a new k, not the one found in part a
Solve the same as a) with 96.6 replaced by the rule of thumb temperature and t=24

 

[stapel]

c) The question makes no sense.

I think the student is being asked to compare the rule of thumb (which has jump discontinuities, but is easy to "do in your head") with the formula (which is continuous, but requires a calculator).

I could be wrong, of course.

Eliz.

 

[Gene]

Hmmmm. If there is a typo and "what value of k" should be "that value value of k" I have to concur. I puzzled about it long enough for you to beat me to an answer. :twisted: