Math Insight

Chemical pollution model exercises

The chemical pollution model page explained how to solve a discrete dynamical system model for a factory polluting a lake in order to determine the exponential decay toward equilibrium. In the following exercises, you can explore similar models.

Exercise 1

For each of the following systems,

1. Compute $W_0$, $W_1$, $W_2$, $W_3$, and $W_4$.
2. Find the equilibrium value of $W_t$ for the systems.
3. Write a solution equation for the system.
4. Compute $W_{100}$.
5. Compute the half-life $T_{1/2}$ of the system.
6. Compute the time $T_{0.02}$ for the system to get 98% of the way to the equilibrium.

The systems:

1. \begin{align*} W_0&=0\\ W_{t+1} &= 1 + 0.2 W_t \end{align*}
2. \begin{align*} W_0&=0\\ W_{t+1} &= 10 + 0.2 W_t \end{align*}
3. \begin{align*} W_0&=0\\ W_{t+1} &= 100 + 0.2 W_t \end{align*}
4. \begin{align*} W_0&=0\\ W_{t+1} &= 10 + 0.1 W_t \end{align*}
5. \begin{align*} W_0&=0\\ W_{t+1} &= 10 + 0.05 W_t \end{align*}
6. \begin{align*} W_0&=0\\ W_{t+1} &= 10 + 0.01 W_t \end{align*}

Exercise 2

For each of the following systems,

1. Compute $W_0$, $W_1$, $W_2$, $W_3$, and $W_4$.
2. Find the equilibrium value of $W_t$ for the systems.
3. Write a solution equation for the system.
4. Compute $W_{100}$.

The systems:

1. \begin{align*} W_0&=0\\ W_{t+1} &= 1 - 0.2 W_t \end{align*}
2. \begin{align*} W_0&=0\\ W_{t+1} &= 10 - 0.2 W_t \end{align*}
3. \begin{align*} W_0&=0\\ W_{t+1} &= 100 - 0.2 W_t \end{align*}
4. \begin{align*} W_0&=0\\ W_{t+1} &= 10 - 0.1 W_t \end{align*}
5. \begin{align*} W_0&=0\\ W_{t+1} &= 10 - 0.05 W_t \end{align*}
6. \begin{align*} W_0&=0\\ W_{t+1} &=10 - 0.01 W_t \end{align*}

Exercise 3

For each of the following systems,

1. Compute $W_0$, $W_1$, $W_2$, $W_3$, and $W_4$.
2. Describe the future terms, $W_5$, $W_6$, $W_7$, $\cdots$.

The systems:

1. \begin{align*} W_0&=0\\ W_{t+1} &= 1 - W_t \end{align*}
2. \begin{align*} W_0&=\frac{1}{2}\\ W_{t+1} &= 1 - W_t \end{align*}
3. \begin{align*} W_0&=0\\ W_{t+1} &= 1 + W_t \end{align*}
4. \begin{align*} W_0&=0\\ W_{t+1} &= 2 + W_t \end{align*}
5. \begin{align*} W_0&=0\\ W_{t+1} &= 1+ 2 W_t \end{align*}
6. \begin{align*} W_0&=-1\\ W_{t+1} &= 1+ 2 W_t \end{align*}

Selected answers

Once you've worked out some of these exercises, you can check your work with the answers to selected problems.