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Homework Statement
You come across a dead body at 3 PM. Its temperature is 83.6 F. 30 minutes later its temperature is 78.6 F. How long was the guy dead before you found him?
Homework Equations
##\frac{dT}{dt}=-k(T-Tair)##
T(0) = 83.6 F
T(0.5) = 78.6 F
The Attempt at a Solution
This is a separable differential equation. Solving it gives me:
##T(t) = Ae^{-kt}+Tair##
However, I don't seem to be able to solve this completely. It feels like I haven't been given enough information (though I'm sure I have...).
At t=0:
##T(0)=83.6=A+Tair##
At t=0.5:
##T(0.5) = 78.6 = Ae^{-0.5k}+Tair##
Trying to solve for k by getting Tair in terms of A:
##78.6=Ae^{-0.5k}+83.6-A##
##Ae^{-0.5k}=A-5##
##e^{-0.5k}=1-\frac{5}{A}##
##-0.5k = ln|1-\frac{5}{A}|##
##k=-2ln|1-\frac{5}{A}|##
##k=ln(1-\frac{5}{A})^{-2}##
Using this in the 2nd equation where t=0.5:
##78.6=A(1-\frac{5}{A})^{-2}+83.6-A##
##A-5=\frac{A}{1-\frac{5}{A}}##
Dividing by A: ##1-\frac{5}{A}=(1-\frac{5}{A})^{-2}##
The last equation is obviously not correct and I'm not sure what else to do. I've tried getting A in terms of Tair, but it doesn't seem to help. I've also tried using ##\frac{dT}{dt}=-k(T-Tair)## in various ways but I can't make anything work there either.