A pond initially contains 1,000,000 gal of water and an unknown amount of an undesirable chemical....

 

Transcribed Image Text:

A pond initially contains 1,000,000 gal of water and an unknown amount of an undesirable chemical. Water containing 0.01 g of this chemical per gallon flows into the pond at a rate of 300 gal/h. The mixture flows out at the same rate, so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond. (a) Let Q(t) be the amount of the chemical in the pond at time t. Write a differential equation for the amount of chemical in the pond at any time. (b) How much of the chemical will be in the pond after a very long time? Does this limiting amount depend on the amount that was present initially? (c) Now assume that the water is initially free of the chemical. Write down an initial value problem for Q(t). (d) Solve the problem in part (c) for Q(t). How much of the chemical is in the pond after 1 year? (e) At the end of 1 year the source of the chemical in the pond is removed; thereafter pure water flows into the pond, and the mixture flows out at the same rate as before. Write down the initial value problem that describes this new situation. (f) Solve the initial value problem in part (e). How much chemical remains in the pond after 1 additional year (2 years from the beginning of the problem)? (g) Using your solution from part (f), determine how long does it take for Q(t) to be reduced to 10 g. A pond initially contains 1,000,000 gal of water and an unknown amount of an undesirable chemical. Water containing 0.01 g of this chemical per gallon flows into the pond at a rate of 300 gal/h. The mixture flows out at the same rate, so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond. (a) Let Q(t) be the amount of the chemical in the pond at time t. Write a differential equation for the amount of chemical in the pond at any time. (b) How much of the chemical will be in the pond after a very long time? Does this limiting amount depend on the amount that was present initially? (c) Now assume that the water is initially free of the chemical. Write down an initial value problem for Q(t). (d) Solve the problem in part (c) for Q(t). How much of the chemical is in the pond after 1 year? (e) At the end of 1 year the source of the chemical in the pond is removed; thereafter pure water flows into the pond, and the mixture flows out at the same rate as before. Write down the initial value problem that describes this new situation. (f) Solve the initial value problem in part (e). How much chemical remains in the pond after 1 additional year (2 years from the beginning of the problem)? (g) Using your solution from part (f), determine how long does it take for Q(t) to be reduced to 10 g.