Posted by Helga on Tuesday, December 13, 2011 at 12:28am.
You are a forensic doctor called to a murder scene. When the victim was discovered, the body temperature was measured and found to be 20°C. You arrive one hour later and find the body temperature at that time to be 18°C. Assuming that the ambient temperature remained constant in that intervening hour, give the police an estimate of the time of death. (Take 37°C as normal body temperature.)

Newton's Law of Cooling  drwls, Tuesday, December 13, 2011 at 1:59am
Newton's law of convective cooling can be written in the form
dT/dt = C(T  Ta)
where C is a constant that involves the heat capacity and cooling coefficient, and Ta is ambient temperature.
The solution to the differential equation is

Newton's Law of Cooling  drwls, Tuesday, December 13, 2011 at 2:00am
Newton's law can be written in the form
dT/dt = C(T  Ta)
where C i

Newton's Law of Cooling  drwls, Tuesday, December 13, 2011 at 2:05am
My answer seems to have been not fully uploaded. Sorry about that. Computer problems here.

Newton's Law of Cooling  Helga, Tuesday, December 13, 2011 at 5:07am
Shame about your computer. Don't want to die wondering ;)

Newton's Law of Cooling  drwls, Tuesday, December 13, 2011 at 8:31am
The solution is
(TTa) = (ToTa)e^Ct
Where To = 37 C and Ta = ambient temperature
Your two data points give you two equations in three unknowns: t (when reported), Ta and C. The time of second measurement is t + 1 hr. More information is needed to make an estimate. You would need the value of ambient temperature or the Newton heat loss coefficient C

Newton's Law of Cooling  Helga, Tuesday, December 13, 2011 at 9:31am
Oh, I see. So to make an estimate you'd have to make an assumption of ambient temperature, right? Assuming constant ambient temperature of 10°C, the time of death could be estimated to be at around 4.45 (i.e 4 to 4 1/2) hours before discovery. Is that correct?

Newton's Law of Cooling  drwls, Tuesday, December 13, 2011 at 12:27pm
I have not done the numbers but that sounds reasonable. Did they tell you to assume ambient was 10 C?
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