Expert: Abe Mantell - 1/4/2009

Question
An aluminum beam was brought from the outside cold into a machine shop where
the temperature was held at 65F. After 10 minutes, the beam warmed to 35F and after
another 10 minutes it was 50F. Use Newton’s law of cooling to estimate the beam’s initial
temperature.

Answer
Newton's Law of Cooling/Heating states that the rate of change of
temperature of an object is proportional to the difference between
the ambient temperature and the temperature of the object.  In
differential form, with T=the temp. of the object, and Ta=ambient temp,
we get: dT/dt = k(T-Ta), where k=constant of proportionality

This is separable: dT/(T-65) = k dt, integrating both sides gives
ln|T-65|=kt+c1, where c1=contsant, solving for T gives
T-65=e^(kt+c1) ==> T=65+(e^kt)(e^c1), but e^c1=C, another constant.
So we get: T=65+Ce^kt, now we impose the initial conditions to solve
for C & k.

When t=10 (10 minutes later), T=35 ==> 35=65+Ce^(10k) -- eq. 1
When t=20 (20 minutes later), T=50 ==> 50=65+Ce^(20k) -- eq. 2

Solving eq. 1 for C, we get: C=-30e^(-10k)
Now substituting that into eq. 2, we get: 50=65+[-30e^(-10k)]e^(20k)
==> -15=-30e^(10k) ==> e^(10k)=1/2 ==> 10k=ln(1/2) ==> k=ln(1/2)/10
==> k=-0.0693147..., which gives C=-60.

Thus, T=65-60e^(-0.069315t) degrees F.

What was the initial temp?  When t=0 ==> T=5 deg. F.

OK?

Abe