Consider a pond that initially contains 10 million gallons of fresh water. Water

Question

Consider a pond that initially contains 10 million gallons of fresh water. Water containing a chemical pollutant flows into the pond at the rate of 5 million gallons per year (gal/yr), and the mixture in the pond flows out at the same rate. The concentration c=c(t) of the chemical in the incoming water varies periodically with time to the expression c(t) = 2 + sin(2t) grams per gallon (g/gal).

Construct a mathematical model of this flow process and determine the amount of chemical in the pond at any time t. Then, plot the solution using Maple and describe in words the effect of the variation in the incoming chemical.

Explanation / Answer

Pond initially contains 10 million gallons of fresh water. Water containing toxic waste flows into pond at rate
of 5 million gal/year, and exits pond at same rate. Concentration is c(t) = 2 + sin 2t g/gal of toxic waste in
incoming water.

Assume toxic waste is neither created or destroyed in pond, and distribution of toxic waste in pond is uniform (stirred). Then

Rate in: (2 + sin 2t g/gal)(5 x 106 gal/year). Rate out: If there is Q(t) g of toxic waste in pond at time t, then concentration of salt is Q(t) lb/107 gal, and it flows out at rate of [Q(t) g/107 gal][5 x 106 gal/year]

Rate in: (2 + sin 2t g/gal)(5 x 106 gal/year)
Rate out: [Q(t) g/107 gal][5 x 106 gal/year] = Q(t)/2 g/yr.
Then initial value problem is

Change of variable (scaling): Let q(t) = Q(t)/106. Then

To solve this initial value problem we use the method of integrating factors:

Using integration by parts (see below) and the initial condition, we obtain after simplifying,

Here is the detail of the integration by parts:

A graph of solution along with direction field for differential equation is given below. Note that exponential term is important for small t, but decays away for large t. Also, y = 20 would be equilibrium solution if not for sin(2t) term.

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