Question

Consider a lake of constant volume 12200 km^3, which at time t contains an amount y(t) tons of pollutant evenly distributed throughout the lake with a concentration y(t)/12200 tons/km^3.
assume that fresh water enters the lake at a rate of 67.1 km^3/yr, and that water leaves the lake at the same rate. suppose that pollutants are added directly to the lake at a constant rate of 550 tons/yr.

A. Write a differential equation for y(t).

B. Solve the differential equation for initial condition y(0)=200000 to get an expression for y(t). Use your solution y(t) to describe in practical terms what happens to the amount of pollutants in the lake as t goes from 0 to infinity.

Answer 1

Assumption: To ignore evaporation and seapage so that all the water leaving the lake does so via a stream, and it does so carrying pollution with it as it goes.

a) Differential equation:

Since y=200000 when t=0, 200000 = 12200/61.7 *(550-C")

b) put t = 0

The pollution starts at 200000 tons and drops toward 100,000 tons as time passes.

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