Time of death problem using differential equation

The problem states: Give an interval that estimates for the person's time of death. Ts= 21.1 degrees C room temperature, Normal temperature of the body Ta= is an interval of 36.6 to 37.2 degrees C. Temperature of the body when found at midnight is Ti= 34.8 degrees C. The final temperature half hour later of the body Tf= 34.3 degrees C.

So using Newton's cooling equation:

K=(T1-T2)=-ln((T1'-s)/(T2'-s))

where the temperature of the corpse is:

K=-(1/2)ln((Tf - Ts)/(Ti - Ts)

Time of death is:

D=-(1/k)ln((Ta - Ts)/(Ti - Ts))

So i'm confused on where to intergate? I wanted to intergate for the time of death for To but in respects to time but there is not a time variable in that equation so I'm stumped. Thanks to all who can help.

Re: Time of death problem using differential equation

Re: Time of death problem using differential equation

Thank you TheEmptySet for your help! You made understanding this problem a lot easier! I have one question though. What do you mean by the initial temperatures A? Is it the intial temperature (Ti) the body was found and final temperature (Tf) or is it the average body temperature (Ta) and the final (Tf)? Sorry I'm sure I'm way over thinking this! If anyone else has the answer please respond.

Re: Time of death problem using differential equation

Re: Time of death problem using differential equation

Ok that makes sense. I confused myself there for a minute. Thank you again for your help!