TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

This exercise encourages students to grapple with the meanings of individual terms in the differential equations.

Questions (2) and (3) guide the students to more general models in which mixing is not assumed to be instantaneous (at least not over a large region). This helps students understand why the original seemingly unreasonable assumption was made --- namely, without the instantaneous mixing assumption, the problem is quite complicated.

Prerequisites

For (1) and (2): Linear 1st order differential equations OR the derivative and a graphical DE Solver tool such as DSolve in Mathematica.

For the entire problem, the students should have been introduced to systems of ordinary differential equations.

Time allotment - time management

This project should take 1 day in class to finish most of (1) and (2); and then a week out of class for (3) and (4).

Expectations

Students should be able to handle the Problem 1, and should also be able to figure out appropriate models for Problem 2 and 3 assuming that they have worked with other models of 1st order differential equations.

Some students may arrive at non-linear models. You may wish to suggest out right that all models should be kept linear. In this case, there really aren't many reasonable choices for models.

Future payoffs

Reinforces the modeling concept for differential equations: The changes of a state variable is equal to the incoming rate minus the outgoing rate.

Reinforces the notion that the equations are models -- not reality, and opens the door to experimentation with better models.

Extensions

References and Sources