# Math Insight

### Applet: Model of chemical pollution in a lake

The chemical waste $W_t$ in a lake $t$ days after a factory opens moves toward the equilibrium $$E= \frac{RV}{F}.$$ The waste level $W_t$ is shown by the red curve and the equilibrium value is shown by the horizontal cyan line. You can adjust the parameters by dragging the green points on the sliders. The rate of decay toward equilibrium is captured by the half-life $T_{1/2}$ and the time $T_{0.02}$ for the waste to get 98% the way to equilibrium (i.e., the time for the deviation from equilibrium to decay to 0.02 its original value). You can zoom in and out with the buttons labeled by arrows.

The waste after $t$ days is determined by the discrete dynamical system \begin{align*} W_0 &=0\notag\\ W_{t+1} &= R+\left(1- \frac{F}{V}\right)W_t, \end{align*} where $V$ is the volume of the lake in m3, $F$ is the daily flow of water through the lake in m3, and $R$ is the daily release of chemicals in kg. The solution is \begin{align*} W_t = E -E \left(1- \frac{F}{V}\right)^t. \end{align*}

Applet file: chemical_pollution_lake_model.ggb