Question

A forensic detective is called to the scene of a murder at 2 am. When she checks the temperature of the body, she sees that it is 80 degrees Fahrenheit. The temperature of the room in which the body is found is 65 degrees. If the detective uses Newton's law of cooling, what will she say is the time of death? T(t)=Ta+(To-Ta)e^(-0.1947t).
Thank you to anyone who helps! :)

Answer 1

Newton's Law of cooling is
$T(t)=T_{a}+(T_{0}-T_{a})e^{-0.1947t}$
where
T(t) = temperature of the body after t hours, 80 °F
$T_{a}$ =  ambient temperature, 65 °F
$T_{0}$ = normal body temperature, assumed to be 98.6 °F

The elapsed time since death is obtained from
$65+(98.6-65)e^{-0.1947t|} = 80$
$33.6e^{-0.1957t}=15$
$-0.1947t=ln \frac{15}{33.6}$
$t= \frac{ln(15/33.6)}{-0.1947} =4.142$

4.142 hours before 2 am is 9:52 pm (the previous day) approximately.

Answer:  9:52 pm the previous day.