# At what time did the murder take place? (Differential Equations)

1. Jun 2, 2009

### IRNB

1. The problem statement, all variables and given/known data

Newton’s law of cooling states that the rate of change of temperature of an object is proportional
to the difference between the temperature, T, of the object and that of its surroundings,
Ts. Derive the solution

T(t) = Ts + (T0 − Ts) e−kt,
where T0 is the temperature at t = 0 and k is a constant whose meaning should be identified.
The corpse was found in an air-conditioned room. The forensic scientist measured the body
temperature of the victim at 2am and found it to be 25C; by 3am it had fallen to 21C. The
temperature of a living body is 37C and the temperature of the room was 19C.
At what time did the murder take place?

(Hints: measure time from the time of death, td; write down equations for the body temperature
at 2am & 3 am.)

2. Relevant equations

for y' + ay = b

y=b/a + Ce-ax

3. The attempt at a solution

Almost at a complete loss on this one.

for the first part of this question i have

T' =k(T - Ts)
T'/k = T-Ts
T'/k - T = -Ts

so using y=b/a + Ce-ax for y'+ay=b

T= Ts + Cekt

I'm not sure this is correct since I'm missing a negative sign before the k in the exponential. also what is k? I know its some kind of proportionality constant but what name does it have? I'm also not sure how to show that C = (T0-Ts).

for the second part i have

25=19+Ce-kt1
21=19+Ce-kt2

Ce-kt1=6
Ce-kt2=2

dividing one by the other gives 3=e-k(t2-t1) -- t2-t1=1hr

also
37=19+Ce-kt0

Ce-kt0=18

dividing this by Ce-kt1=6 gives

e-k(t1-t0)=3 where t0 is the time of death.

since e-k(t1-t0)=e-k(t2-t1)=3
t1-t0 must = t2-t1 = 1hr

therefore the murder must have taken place one hour before the first temperature reading was taken at 2am. so the murder must have taken place at 1am.
this seems correct to me but my lecturer seems to think it took place at 12am. Can anyone see where i might have gone wrong or is this one of those rare occasions when the lecturer is incorrect?

2. Jun 2, 2009

3. Jun 2, 2009

### LowlyPion

Roughing it out I get:

6 = 18/ekt1

ekt = 3 or kt = 1.1

2 = 18/ek(t1 + 1)

ek(t1 + 1) = 9 or k*(t1 + 1) = 2.2

==> k = 1.1 and t = 1

Same result. Perhaps a polite inquiry of your lecturer ...?

4. Jun 2, 2009

### IRNB

Thanks guys. Perhaps my lecturer was just in hurry or something and made a mistake.

can anyone help out with the first part of the question? does the negative sign simply mean that the temperature is decreasing and the rate at which it changes also decreases with time? also does anyone have any ideas on how to show that C=T0-Ts?

Thanks again guys.

5. Jun 2, 2009

6. Jun 2, 2009

### IRNB

thanks a lot LowlyPion that did help.

this case is now closed. :P