|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
78.5o 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 (98.6o 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,
= Te + (T0 − Te ) e - kt,
where T (t) is
the temperature of the object at time t, Te 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 80o F.
One hour later, τ + 1 hours after
death, the body was found to be 78.5o F.
Our known constants for this problem are, Te = 68o F
and T0 = 98.6o F.
At what time did our victim die?
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Biology Project > Biomath > Applications > Cooling
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The Biology Project
Department of Biochemistry and Molecular Biophysics
The University of Arizona
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