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 *T*_{e} are the temperatures of
the object and its environment respectively, then

where

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

where

- 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.