A lake, with volume V = 100 km3, is fed by a river at a rate of r km3/yr. In addition, the...

A lake, with volume V = 100 km3, is fed by a river at a rate of r km3/yr. In addition, there is a factory on the lake that introduces a pollutant into the lake at the rate of p km3/yr. There is another river that is fed by the lake at a rate that keeps the volume of the lake constant. This means that the rate of flow from the lake into the outlet river is (p + r) km3/yr. Let x(t) denote the volume of the pollutant in the lake at time t. Then c(t) = x(t)/ V is the concentration of the pollutant.

(a) Show that, under the assumption of immediate and perfect mixing of the pollutant into the lake water, the concentration satisfies the differential equation

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(b) It has been determined that a concentration of over 2% is hazardous for the fish in the lake. Suppose that r = 50 km3/yr, p = 2 km3/yr, and the initial concentration of pollutant in the lake is zero. How long will it take the lake to become hazardous to the health of the fish?