Solving a Chemical Distribution Problem in a Pond Using Differential Equations

[ehabmozart]

1. Homework Statement

Apond 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) Write a differential equation for the amount of chemical in the pond at any time.

Homework Equations

This should be done without a prerequisite of any formula

The Attempt at a Solution

Well, I do have the solution which says let q(t) be the amount chemical at any time. Hence, the concentration of this chemical in the water at any time is q(t) / 1000000. (This part is understood). 300*(0.01) is the amount of chemical coming into the pond every hour. (This part is fine too).. Now, the last and most important part ... 300 q(t) / 100000 is the rate at which the chemical leaves the pond per hour. What?? How did they give this claim?? I mean, in the text it is written that " The mixture flows out at the same rate, so the amount of water in the pond
remains constant. ". I am completely perplexed here. Honestly, i need to know some basics of DE. If anyone would give me a hand on how to approach the answer, I would owe him/her big time. Thanks a lot for your time!

 

[pasmith]

1. Homework Statement

Apond 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) Write a differential equation for the amount of chemical in the pond at any time.

Homework Equations

This should be done without a prerequisite of any formula

The Attempt at a Solution

Well, I do have the solution which says let q(t) be the amount chemical at any time. Hence, the concentration of this chemical in the water at any time is q(t) / 1000000. (This part is understood). 300*(0.01) is the amount of chemical coming into the pond every hour. (This part is fine too).. Now, the last and most important part ... 300 q(t) / 100000 is the rate at which the chemical leaves the pond per hour. What?? How did they give this claim?? I mean, in the text it is written that " The mixture flows out at the same rate, so the amount of water in the pond
remains constant. ". I am completely perplexed here. Honestly, i need to know some basics of DE. If anyone would give me a hand on how to approach the answer, I would owe him/her big time. Thanks a lot for your time!

"The mixture flows out at the same rate, so the amount of water in the pond remains constant" means that the rate at which water flows out is the same as the rate at which it flows in, ie. 300 gallons per hour. But it is nowhere said that the chemical leaves at the same concentration as it enters. Instead the assumption is that the chemical is uniformly distributed throughout the pond, so that concentration at the outflow is the same as it is everywhere in the pond, ie. $q(t)/10^6$. The rate at which it leaves is then $300q(t)/10^6$.