Newton's law of cooling and more!

A detective is called to the scene of a crime where a dead body has just been found. She arrives on the scene at 10:23 pm and begins her investigation. Immediately, the temperature of the body is taken and is found to be 80o F. The detective checks the programmable thermostat and finds that the room has been kept at a constant 68o F for the past 3 days.

After evidence from the crime scene is collected, the temperature of the body is taken once more and found to be 78.5^o F. This last temperature reading was taken exactly one hour after the first one. The next day the detective is asked by another investigator, “What time did our victim die?” Assuming that the victim’s body temperature was normal (98.6^o F) prior to death, what is her answer to this question? Newton's Law of Cooling can be used to determine a victim's time of death! Cool, or what!

So, what exactly is Newton's Law Of Cooling?
Newton’s Law of Cooling describes the cooling of a warmer object to the cooler temperature of the environment. Specifically we write this law as,

T (t) = T_e + (T₀ − T_e ) e ^{- kt},

where T (t) is the temperature of the object at time t, T_e is the constant temperature of the environment, T₀ is the initial temperature of the object, and k is a constant that depends on the material properties of the object.

To organize our thinking about this problem, let’s be explicit about what we are trying to solve for. We would like to know the time at which a person died. In particular, we know the investigator arrived on the scene at 10:23 pm, which we will call τ hours after death. At 10:23 (i.e. τ hours after death), the temperature of the body was found to be 80^o F. One hour later, τ + 1 hours after death, the body was found to be 78.5^o F. Our known constants for this problem are, T_e = 68^o F and T₀ = 98.6^o F.
At what time did our victim die?

Using Newton’s Law of Cooling we have the following two equations,
T ( τ ) = 80 = 68 + (98.6 − 68) e - kτ ,	.....(1)
T ( τ + 1) = 78.5 = 68 + (98.6 − 68) e - k ( τ + 1),	.....(2)
where τ corresponds to 10:23 and represents the time (in hours) since death.
Simplifying (1) yields,
12 = 30.6 e - kτ ,	.....(3)
Using the laws of exponents and simplifying (2) yields,
10.5 = 30.6 e - kτ e − k.	......(4)
Notice that in (3) and (4), we have two equations and two unknowns (k and τ). Solving for e - kτ in (3) gives,

Substituting this value of e ^{- kτ} in (4) gives,

Now that we have a value for k, we can use this to solve for the remaining unknown, τ. Substituting k ≈ 0.134 into (3) yields,

We have found τ ≈ 7 hours. Since the detective arrived on scene at 10:23 pm (τ hours after death), the individual must have died 7 hours prior to 10:23 pm or at approximately 3:23 pm.

Thus, the detective would confidently answer the other investigator, “The time of death was approximately 3:23 pm.”
Bingo!