Evergreen Pond contains 100,000 gallons of water. Suppose that
Sunny Stream starts, at some time, to flow into the pond at 100
gallons per hour, with 100 gallons per hour of overflow (out of the
pond) through Buttermilk Falls, so that the amount of water in the
pond stays constant.
Suppose further that:
Sunny Stream contains 1/100 grams/gallon of an important mineral M.
Initially the pond contains a negligible amount of the mineral.
The currents in the pond keep it "evenly mixed" so that
concentrations of substances in the pond are equal everywhere in the
pond.
The concentration of the mineral M in the pond must reach a level
of 10 grams per gallon for extraction to become cost-effective.
Let M(t) be measure the number of grams of the mineral M in the
pond, at time t.
- Set up a differential equation which models the amount
of mineral M in the pond at time t.
- Solve the differential equation for M(t).
- Will the concentration of the mineral in the pond reach
a cost-effective concentration? If so, when? If not, is there an upper
bound for the attainable concentration?