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Supplementary problem for Chapter 1

This problem can be solved with your bare hands, but you may wish to use a calculator or computer and/or an integral table.

Recall that Newton's Law of Cooling asserts that the rate that heat is transferred between an object and its environment is proportional to the difference between the temperature of the object and the temperature of its environment. In other words, if T and Te are the temperatures of the object and its environment respectively, then

\begin{displaymath}\frac{dT}{dt} = -k (T-T_e)
\end{displaymath}

where k is a constant of proportionality which depends on how well the object transfers heat, and t measures time.

For a certain well-insulated house, it's observed that k=3 when temperature is measured in degrees centigrade and time is measured in days. The January temperature is well approximated by

\begin{displaymath}T_e = 5 \cos(2 \pi t)
\end{displaymath}

where t=0 corresponds to high noon on January 1.

1.
Use Newton's Law of Cooling to come up with a differential equation for the temperature T inside the house.
2.
Solve the differential equation for T, and give a convincing argument that the temperature in the house will eventually settle in to a periodic oscillation that does not depend on the initial temperature in the house.
3.
Estimate the minimum indoor temperature (after the transient effects of the initial temperature dissipate), and the time of day when it is achieved.


 
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Frank Beatrous
1998-10-03