# A body was found at 10 a.m. in a warehouse where the temperature was 40°F. The medical examiner found the temperature of the body to be 80°F. What was the approximate time of death?

## Use Newton's law of cooling with k=0.1947.

Aug 9, 2016

Approximate time of death is $8 : 02 : 24$ am.

Important to note that this is the skin temperature of the body. The medical examiner would be measuring internal temperature which would decrease much slower.

#### Explanation:

Newton's law of cooling states that the rate of change of temperature is proportional to the difference to the ambient temperature. Ie

$\frac{\mathrm{dT}}{\mathrm{dt}} \propto T - {T}_{0}$

If $T > {T}_{0}$ then the body should cool so the derivative should be negative, hence we insert the proportionality constant and arrive at

$\frac{\mathrm{dT}}{\mathrm{dt}} = - k \left(T - {T}_{0}\right)$

Multiplying out the bracket and shifting stuff about gets us:

$\frac{\mathrm{dT}}{\mathrm{dt}} + k T = k {T}_{0}$

Can now use the integrating factor method of solving ODEs.

$I \left(x\right) = {e}^{\int k \mathrm{dt}} = {e}^{k t}$

Multiply both sides by $I \left(x\right)$ to get

${e}^{k t} \frac{\mathrm{dT}}{\mathrm{dt}} + {e}^{k t} k T = {e}^{k t} k {T}_{0}$

Notice that by using the product rule we can rewrite the LHS, leaving:

$\frac{d}{\mathrm{dt}} \left[T {e}^{k t}\right] = {e}^{k t} k {T}_{0}$

Integrate both sides wrt to $t$.

$T {e}^{k t} = k {T}_{0} \int {e}^{k t} \mathrm{dt}$

$T {e}^{k t} = {T}_{0} {e}^{k t} + C$

Divide by ${e}^{k t}$

$T \left(t\right) = {T}_{0} + C {e}^{- k t}$

Average human body temperature is 98.6°"F".

$\implies T \left(0\right) = 98.6$

$98.6 = 40 + C {e}^{0}$

$\implies C = 58.6$

Let ${t}_{f}$ be the time at which body is found.

$T \left({t}_{f}\right) = 80$

$80 = 40 + 58.6 {e}^{- k {t}_{f}}$

$\frac{40}{58.6} = {e}^{- k {t}_{f}}$

$\ln \left(\frac{40}{58.6}\right) = - k {t}_{f}$

${t}_{f} = - \ln \frac{\frac{40}{58.6}}{k}$

${t}_{f} = - \ln \frac{\frac{40}{58.6}}{0.1947}$

${t}_{f} = 1.96 h r$

So from time of death, assuming body immediately started to cool, it took 1.96 hours to reach 80°F at which point it was found.

$1.96 h r = 117.6 \min$

Approximate time of death is $8 : 02 : 24$ am