Newton's Law of Cooling with no Ambient Temperature

[TobyKenz1]

This is a 4 part ordinary differential equation problem.

a.) You are a member of a CSI team that has discovered a dead body in a field. Upon discovery of the body you measure its temperature and find that it is 76.04058181\circF. 10 minutes later the temperature of the body is 73.42926519\circF. 10 minutes after that the temperature of the body is 71.06644821\circF. SET UP BUT DO NOT SOLVE the differential equation(s) and condition(s) to determine the temperature, T(t), of the body at any time t.

b.) The solution of your differential equation is T(t)=A+50e^{-0.6}e^{-0.01t, where A is a constant. Find A.

c.) What is the outside temperature in the field?

d.)When did the individual expire (normal body temperature is 98.6\circF)?

For the first part, I determined all the conditions that were give, and I wrote the equation

dT/dt=k(T-Ta)

where Ta is the ambient temperature. For the second part, I got A=27.441 by using the original temperature, 76.04058181 at time equals 0.

I just don't know how to go about the third and fourth parts. Any help would be very much appreciated.

 

[TobyKenz1]

Correction: I got A=48.6, using the original temperature at time 0.

 

[TobyKenz1]

Correction2: Would it hold to reason for c that since the limit of the temperature function is 48.6 as t approaches infinity that 48.6 must be the ambient temperature? Because I think I was massively overthinking this problem.

This would also make part D much easier...