Crime Scene

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.

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?

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