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.
Rivers do not flow at the same rate the year around. They tend to be full in the Spring when the snow melts, and to flow more slowly in the Fall. To take this into account, suppose the flow of our river is
ri = 50 + 20cos(2π(t − 1/3)).
Our river flows at its maximum rate one-third into the year, i.e., around the first of April, and at its minimum around the first of October.
a) Setting p = 2, and using this flow rate, use dfield6 to plot the concentration for several choices of initial concentration between 0% and 4%. (If your solution seems erratic, reduce the relative error tolerance using Options→Solver settings.) How would you describe the behavior of the concentration for large values of time?
b) It might be expected that after settling into a steady state, the concentration would be greatest when the flow was smallest, around the first of October. At what time of the year does the highest concentration actually occur? Reduce the error tolerance until you get a solution curve smooth enough to make an estimate.
We need at least 10 more requests to produce the solution.
0 / 10 have requested this problem solution
The more requests, the faster the answer.