From experimental observations it is known that (up to a
``satisfactory'' approximation) the surface temperature of an object
changes at a rate proportional to its relative temperature. That is,
the difference between its temperature and the temperature of the
surrounding environment. This is what is known as **Newton's law of
cooling**. Thus, if is the temperature of the object at time
*t*, then we have

where *S* is the temperature of the surrounding environment. A
qualitative study of this phenomena will show that *k* >0. This is
a first order linear differential equation. The solution, under the
initial condition , is given by

Hence,

,

which implies

This equation makes it possible to find *k* if the interval of time
is known and vice-versa.

**Example: Time of Death** Suppose that a corpse
was discovered in a motel room at midnight and its temperature was
. The temperature of the room is kept constant at
. Two hours later the temperature of the corpse dropped to
. Find the time of death.

**Solution:** First we use the observed
temperatures of the corpse to find the constant *k*. We have

.

In order to find the time of death we need to remember that the temperature of a corpse at time of death is (assuming the dead person was not sick!). Then we have

which means that the death happened around 7:26 P.M.

One of our interested readers, E.P. Esterle, wrote a program that helps find the time of death based on the above notes. Click HERE to download it. Have fun with it.

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