Timothy A.

Differential Equations

1 year, 2 months ago

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 pollutant enters the lake at a rate $r,$ and that water leaves the lake at the same rate. Suppose that pollutants are also added directly to the lake at a constant rate $P$. Note that the given assumptions neglect a number of factors that may, in some cases, be important; for example, the water added or lost by precipitation, 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.
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\begin{array}{l}{\text { (a) If at time } t=0 \text { the concentration of pollutant is } c_{0} \text { , find an expression for the concen- }} \\ {\text { tration } c(t) \text { any time. What is the limiting concentration as } t \rightarrow \infty \text { ? }} \\ {\text { (b) If the addition of pollutants to the lake is terminated }(k=0 \text { and } P=0 \text { for } t>0\text { ), } } \\ {\text { determine the time interval } T \text { that must elapse before the concentration of pollutants is }} \\ {\text { reduced to } 50 \% \text { of its original value; to } 10 \% \text { of its original value. }} \\ {\text { (c) Table } 2.3 .2 \text { contains data for several of the Great Lakes. Using these data determine }}\end{array}
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$$
\begin{array}{l}{\text { from part (b) the time } T \text { necessury to reduce the contamination of each of these lakes to }} \\ {10 \% \text { of the original value. }}\end{array}
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