Mathematics & Science Learning Center
Computer Laboratory

Applications of Differential Equations

Compartmental Analysis

Exercises For The Student

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Problem 2

Lake Whiffy has suffered from pollution emanating from the Stink Inc. chemical plant on its shoreline for the last fifty years. As a result, the waters of the lake—a total of 50,000,000 gallons—currently contain a total of 1,000,000 pounds of toxic waste. The Environmental Protection Agency has just told Stink Inc. that they must reduce the amount of toxic waste that they dump into the lake to a total of 4 pounds per day. The total amount of liquid entering Lake Whiffy from streams and rainfall, and including the waste dumped by Stink Inc., is 20,000 gallons a day, and this same amount per day leaves the lake by the River Ooze, so that the lake level is always constant.

Conduct the following exercises:

1. Write down the initial value problem that models this scenario. (You should probably do this step on paper, not with Mathematica.)

2. Use Mathematica to solve the initial value problem that you just wrote down, reading your result into ex2.

3. Use the [[1,1,2]] trick to strip out the core of the solution Mathematica returns, reading the result back into ex2.

4. Plot the graph of ex1 on the time interval covering the next 50 years. (Remember to convert this time into days, and assume that there are 365 days in a year.) Read the resulting graph into the variable ex2plot.

5. After 50 years have passed, what amount of toxic waste will remain in the lake?

6. Looking at your graph, what is the long term behavior of your solution? Does this make sense? Confirm your thinking by using the Limit command, letting t→∞.

Let's go and see what you should have gotten...

If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.