Rivers do not flow at the same rate the year around. They tend to be full in the spring...

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 the input river in Exercise 9 is

r = 50 + 20 cos(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 your numerical solver to plot the concentration for several choices of initial concentration between 0% and 4%. (You might have to reduce the relative error tolerance of your solver, perhaps to .) How would you describe the behaviour 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 (i.e., around the first of October). At what time of year does it actually occur?

Reference: Exercise 9:

A lake, with volume is fed by a river at a rate of In addition, there is a factory on the lake that introduces a pollutant into the lake at the rate of 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 Let x (t) denote the volume of the pollutant in the lake at time t. Then t(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

(b) It has been determined that a concentration of over 2% is hazardous for the fish in the lake. Suppose that 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?