#### Modeling the Pollution in a Lake Problem

Most of the water flowing into Lake Ontario is from Lake Erie. Suppose that pollution of the lakes ceased, except for pollution from an aluminum factory on Lake Ontario. How long would it take for the pollution level in each lake to be reduced to 10 percent of its present level?

First, to simplify matters, let's assume that 100 percent of the water in Lake Ontario comes from Lake Erie. Let $a(n)$ and $b(n)$ be the total amount of pollution in Lake Erie and Lake Ontario, respectively, after $n$ years. Since pollution has stopped, the concentration of pollution in the water coming into Lake Erie is $c = 0$. It has also been determined that, each year, the percentage of water replaced in Lakes Erie and Ontario is approximately 38 and 13 percent, respectively. Additionally, suppose that an aluminum factory on Lake Ontario directly dumps 25 units of pollutant into the lake each year. Initially, there are 2500 units of pollutant in Lake Ontario, and 3150 units of pollution in the lake after 1 year.

**Problem:**

1. Write a discrete dynamical system, using a system of difference equations, which models this process, and convert these to a second order discrete dynamical system that will tell us the total amount of pollution in Lake Ontario.

2. Find the general solution to this discrete dynamical system.

3. Does this system have an equilibrium value? Justify your answer.

4. Find the particular solution and determine how long it would take for the pollution level in Lake Ontario to be reduced to 10 percent of its present level.

5. Describe the long term behavior of this system.