Differential Equations Question - Temperature

Here’s the problem:

A body was found dead in the refrigerator at 5:30am at 6 am the coroner measured the core temp of the body to be 85 degrees at 6:30 it was measured again, 84 degrees. Assume the body was killed in the refrigerator what time was the person killed?

I figured this part out. I used:

dT/dt = k(T-Tm)

Solving this:

T(t) = (T-Tm)e^kt +Tm

T is the temperature at time t and Tm is the temp around the body, the environmental temperature.

I solved this and got t = 5.3.

We needed to use positive numbers for t so the book assigned 6:00 am to t = 0 and so t= 1 would be 5 am and t = 5.3 would be roughly 12:42am.

Before I go to the next part I understand you can solve this with something called Laplace Transform but I haven’t learned that yet.

That part I think I understand. Now assume that the person wasn’t killed in the refrigerator and let h represent the number of hours in the refrigerator. So at h = 1 the body was moved in to the refrigerator at 5:00am what time was the person killed? And so on with h = 1 then 2 then 3 …

Filling in this chart:

h time body moved time of death.
12 6:00pm
11
10
9
8
7
6
5
4
3
2
1

I think my trouble is this idea of ‘h’
Numbers that need to be known are

Tm in the refrigerator is 50 degrees F
Tm in the diner is 70 degrees F
The person was initially 98.6 degrees
In the beginning I solved for e^k which I got 1.06

So the equations I’m working with is:
In refrigerator
T(t) = (T-50)(1.06)^h + 50 (*)

In the diner:
T(t) = (T-70)(1.06)^t + 70 (**)

I thought I needed to work backwards

If he was in the refrigerator for h= 1 plug this into (*) but I get 83 which doesn’t make sense because he needed to be warmer when he went into the refrigerator to be cooled down to 85. I was looking for that number to be 88 or 87 ish then I could use that as the target temp for (**) to solve for the time to get to 88 or 87 ish.

If you could be any help that would be great. Thanks
 

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