In Exercise we will consider a certain lake which has a volume of V = 100 km3. It is fed by a river at a rate of ri km3/year, and there is another river which is fed by the lake at a rate which keeps the volume of the lake constant. In addition, there is a factory on the lake which introduces a pollutant into the lake at the rate of p km3/year. This means that the rate of flow from the lake into the outlet river is (p + ri) km3/year. Let x(t) denote the volume of the pollutant in the lake at time t, and let c(t) = x(t)/V denote the concentration of the pollutant.
Show that, under the assumption of immediate and perfect mixing of the pollutant into the lake water, the concentration satisfies the differential equation c′ + ((p + ri)/V)c = p/V.
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