Problems 45 and 46 deal with a shallow reservoir that has a one-square-kilometer water surface and an average water depth of 2 meters. Initially it is filled with fresh water, but at time t = 0 water contaminated with a liquid pollutant begins flowing into the reservoir at the rate of 200 thousand cubic meters per month. The well-mixed water in the reservoir flows out at the same rate. Your first task is to find the amount x(t) of pollutant (in millions of liters) in the reservoir after t months.

The incoming water has a pollutant concentration of c(t) = 10 liters per cubic meter (L/m³). Verify that the graph of x(t) resembles the steadily rising curve in Fig. 1.5.9, which approaches asymptotically the graph of the equilibrium solution x(t) ≡ 20 that corresponds to the reservoir’s long-term pollutant content. How long does it take the pollutant concentration in the reservoir to reach 10 L/m³?

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25+x 20 15 10 5 x = 20 ^^^^^^ Problem 46 Problem 45 30 40 50 60 Graphs of solutions in 10 20 FIGURE 1.5.9. Problems 45 and 46.