- #1

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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.