# Homework Help: Extreme Newton's Law of Cooling D.E.

1. Apr 27, 2012

### bdh2991

1. The problem statement, all variables and given/known data
after a very unpleasant valentine's Day, a dead body was found in a downtown warehouse that had no heating or air conditioning. it was February in Florida and we know that the daily temperature in the warehouse fluctuates according to the function T(t)= 63-12sin(∏t/12), where t=0 corresponds to midnight on any given day. The body was discovered at 1:30 am on Feb. 15 and its temperature was 73 degrees F. Two hours later the temperature was 68 degrees F. What was the time of death of this body?

2. Relevant equations

dT/dt = k (T - T(m))

[exp(at)(-BcosBt + asinBt)]/a^2+b^2

where k is the proportionality constant T is temperature and T(m) is the medium of the environment surrounding the object.

3. The attempt at a solution

i set up my differential as dT/dt - kT - 63k = 12ksin(∏t/12)

after doing the integrating factor method and using the equation to solve the integration i got some huge formula for T

then i rescaled the time so i could solve for C and k ... for C i got

C= 73+ (144∏k)/(144k^2+∏^2)

after that i tried to solve for k but it honestly looks impossible and i'm not sure how to do it
if someone could help me out or at least check my work so far i would greatly appreciate it....this problem seems close to impossible

2. Apr 27, 2012

### sid9221

Not a 100% sure as I've never solved one with a separate function for ambient air temperature but maybe you need to sub T(t)=63-12sin(∏t/12) into newtons formula rather than equal to it.

I think the formula should be dT/dt = k [T - {63-12sin(∏t/12)}] where T is the initial temperature given by T(0) = 63.

So dT/dt = k(12sin(∏t/12))

and solve that ?

Last edited: Apr 27, 2012
3. Apr 27, 2012

### HallsofIvy

You should not separate the "63k" and "$12ksin(\pi t/12)$":
$$dT/dt- kT= 63k+ 12ksin(\pi t/12)$$

The associated homogeneous equation, dT/dt-kT= 0, has solution $T= Ae^{kt}$. Now, look for a solution of the form $B+ Csin(\pi t/12)+ Dcos(\pi t/12)$. Put that into the equation and solve for B, C, and D. Then put the entire solution into y(1.5)= 73 and y(3.5)= 68 to solve for A and k.

4. Apr 27, 2012

### bdh2991

ok i see, i'm going to redo it that way, no wonder i was getting nowhere with the way i did it, thanks!