# Math Insight

### Chemical pollution worksheet

Elementary dynamical systems
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1. Consider the dynamical system \begin{align*} W_0&=0\\ W_{t+1} &= 10 + 0.2 W_t. \end{align*}
1. Compute $W_0$, $W_1$, $W_2$, $W_3$, and $W_4$.
2. Find the equilibrium value of $W_t$ for the system. Cobweb the dynamical system for initial conditions near the equilibrium to determine its stability.
3. Calculate the solution for the system. Use this solution to calculate $W_{100}$.
4. Compute the half-life $T_{1/2}$ of the system.
5. Compute the time $T_{0.02}$ for the system to get 98% of the way to the equilibrium.

2. Consider the system \begin{align*} W_0&=0\\ W_{t+1} &= 10 - 0.2 W_t. \end{align*}
1. Compute $W_0$, $W_1$, $W_2$, $W_3$, and $W_4$.
2. Find the equilibrium value of $W_t$ for the system. Cobweb the dynamical system for initial conditions near the equilibrium to determine its stability.
3. Write a solution equation for the system and use it to compute $W_{100}$.

3. For the dynamical system \begin{align*} W_{t+1}-W_t &= 10 - 0.8 W_t\\ W_0=0, \end{align*} find the equilibria and solve the system.