# Differential equations and law of cooling

1. Oct 27, 2013

### fogvajarash

1. The problem statement, all variables and given/known data
A homicide victim is found to have a temperature of 31°C at the stroke of midnight. At 1:00AM his temperature dropped to 29°C. Assuming that the temperature of the room stays at 20°C, when did the murder take place?

2. Relevant equations
-

3. The attempt at a solution
This is more of a problem in how to solve for the constants and variables instead of setting up the problem. By noticing that:

dT/dt = -k(T-N) (where N is the lowest temperature value and T the temperature)

We obtain a solution to the differential equation which is shown below:

T(t) = 20 + (T0 - 20)e-kt

And with this result, we can label the time 12.00AM as h and 1.00AM as h+1:

T(h) = 20 + (T0-20)e-kt = 31
T(h+1) = 20 + (T0-20)e-k(t+1) = 29

With clearing these two expressions (dividing them), we obtain that the value of the constant k is:

k ≈ 0.2007

However, now I can't work out anything else with having the value of the constant. I can't clear neither for T0 (the temperature at the beginning) nor k (the constant). I've dealt with a similar problem but in this case the initial temperature was given so the problem could be worked out. However, how can we proceed in this case?

2. Oct 28, 2013

### Simon Bridge

With the constant k, you now have an equation for the temperature as a function of time.
What is the temperature of a freshly dead person?

3. Oct 28, 2013

### fogvajarash

I'm guessing 37.5 haha. Would this be fair enough? Now i can find To.

4. Oct 28, 2013

### Simon Bridge

You could just take a wild guess .... or you could look up "human body temperature".

IRL: problems do not come with every single value you need written down for you - you have to learn to go look for the them.

5. Oct 28, 2013

### LCKurtz

Isn't that pretty close for $^\circ C$?

6. Oct 29, 2013

### fogvajarash

Thank you very much! I never noticed that we could just have assumed the initial value of the temperature (I completely forgot about that fact).

7. Oct 29, 2013

### Simon Bridge

... yes probably - no units were given and nor was the origin of the number provided.

Human mean body temp is usually quoted as 98.6F or 37.0C.
http://www.nlm.nih.gov/medlineplus/ency/article/001982.htm
http://hypertextbook.com/facts/LenaWong.shtml

This is where you have to show your working to get the marks (or provide a citation if you want to get published.) so I think it is a fair criticism.
It could just have been pulled out of the air and accidentally come close in some units.

No worries - technically you should provide a reason for your assumption.
i.e. if the person died of a fever, you would not assume normal body temp.
You could also have used the range of body-temps, and ended up with a range of time-of-death values - just like they do on CSI :)