Mathematical modeling, like verbal modeling (describing something with words), is a progressive enterprise -- one begins with a rough description

and then adds more detail

The added detail may or may not change the first description dramatically.

In this module we continue to look at an example that triggered a vigorous discussion at a recent meeting at Carroll College that included authors of several leading calculus textbooks. The vigorous part of the discussion began when Ross Finney (Thomas and Finney) presented a slide showing what happened in the walls of a building as the exterior temperature varied. He said that adobe houses were commonly built using bricks whose thickness resulted in a twelve hour lag between the oscillations of the exterior temperature and the oscillations of the interior temperature. A well-known mathematician in the audience leapt to his feet to object that this was impossible. Drawing on his knowledge of Newton's model of cooling with an oscillating ambient temperature --

-- he remarked that the oscillations of the interior temperature could not lag more than six hours behind the oscillations of the exterior temperature. We have studied this equation extensively using graphical, numeric, and symbolic methods and we have seen that the oscillations of the temperature T(t) lag behind the oscillations of the ambient temperature A(t) by at most one-fourth of the period. So these remarks apparently agree with our experience so far. We want to investigate this situation a bit further.

One person suggested that instead of using Newton's model of cooling using two temperatures -- one for the inside of the house and one for the outside of the house -- we should think about three temperatures -- the exterior temperature, the interior temperature, and the temperature inside the walls.

This would lead to a more complex and more realistic model that could be written

• The first of these equations describes the variation of the exterior temperature. Notice that the period of the oscillation is 24 hours.

• The third equation describes the change in the interior temperature using Newton's model of cooling. The constant k₃ depends, in part, on the interior of the house. For a small empty house, k₃ will be relatively large because the temperature of the house will change quickly. For a larger house with lots of furniture, k₃ will be smaller. More precisely, the ratio of the volume of the interior to the area of its walls is the important quantity rather than the size of the interior.

• The second equation is the most interesting. It has two terms. The first term uses Newton's model of cooling to model the heat transfer from the exterior to the brick. The second term uses Newton's model of cooling to model the heat transfer from the interior of the house to the brick. The values of the constants k₁ and k₂ depend on the shape of the brick and its composition. These two constants may be the same because the inside of the house is filled with the same material, air, as the outside of the house. However, if the air in the interior of the house was, for example, more humid then the air outside the house then these constants might be different.

Your CAS window has an example of one numerical solution to a system of equations like the equations above. Experiment using your CAS window with this model. You may want to modify your answer from the previous module in view of your experimental results.

Another person at the Carroll College meeting pointed out that Newton's model of cooling was used to model a situation where one object whose temperature is given by a function A(t) is in contact with another object whose temperature is given by a function T(t) but that this model is at best a crude simplification of what is happening inside the walls of an adobe house. Within the walls the temperature varies. For example, if the exterior temperature was very high and the interior temperature was very low then the temperature of the side of the brick on the outside would most likely be quite warm and the temperature of the side of the brick on the inside would most likely be quite a bit cooler.

She went on to suggest that a better model could be obtained by mentally dividing the brick into two thinner subbricks and keeping track of the temperature within each subbrick

This situation might be described by the differential equations

Another person at the Carroll College meeting immediately noted that an even better model would use even more subbricks.

• Explain each of the differential equations in the two subbrick model.

• Write down a four subbrick model and explain each of your differential equations.

• Based on your work so far, which side would you take in the debate that occurred at Carroll College. Justify your position.