Chapter 2.3, Problem 21P

Consider a lake of constant volume $V$ containing at time $t$ an amount $Q(t)$ of pollutant, evenly distributed throughout the lake with a concentration $c(t)$, where $c(t)=Q(t)/V$. Assume that water containing a concentration $k$ of pollutantenters the lake at a rate $r$, and that water leaves the lakeat the same rate. Suppose that pollutants are also added directly to the lake at a constant rate $P$. Note that the givenassumptions neglect a number of factors that may, in somecases, be important—for example, the water added or lost byprecipitation, absorption, and evaporation; the stratifying effect of temperature differences in a deep lake; the tendency of irregularities in the coastline to produce sheltered bays;and the fact that pollutants are not deposited evenly throughout the lake but (usually) at isolated points around its periphery. The results below must be interpreted in the light of the neglect of such factors as these.

If at time $t=0$ the concentration of pollutant is $c_0$, find an expression for the concentration $c(t)$ at any time. What is the limiting concentration as $t\to \infty$.

If the addition of pollutants to the lake is terminated ( $k=0$ and $P=0$ for $t>0$), determine the time interval $T$ that must elapse before the concentration of pollutants is reduced to $50\%$ of its original value; to $10\%$ of its original value.

Table $\mathrm{2.3.1}$ containsdata for several of the Great Lakes. Using these data, determine from part (b) the time $T$ necessary to reduce the contamination of each of these lakes to $10\%$ of the original value.

TABLE $\mathrm{2.3.1}$ Volume and flow data for the Great Lakes.

$\begin{array}{ccc}\text{Lake}& \text{V}\left({\text{km}}^{\text{3}}{\text{×10}}^{\text{3}}\right)& \text{r}\left({\text{km}}^{\text{3}}\text{/year}\right)\\ \text{Superior}& 12.2& 65.2\\ \text{Michigan}& 4.9& 158\\ \text{Erie}& 0.46& 175\\ \text{Ontario}& 1.6& 209\end{array}$