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.
Suppose that ri = 50, and p = 2.
a) Assume that the factory starts operating at time t = 0, so that the initial concentration is 0. Use dfield6 to plot the solution. Remember the definition of the concentration is x/V so you can be sure it is pretty small. Choose the dimensions of the display window carefully.
b) It has been determined that a concentration of over 2% is hazardous for the fish in the lake. Approximately how long will it take until this concentration is reached? You can “zoom in” on the dfield6 plot to enable a more accurate estimate.
c) What is the limiting concentration? About how long does it take for the concentration to reach a concentration of 3.5%?
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