Please solve allvthe parts of the Question A pond containing 10^6 gal of water i
ID: 3161420 • Letter: P
Question
Please solve allvthe parts of the Question A pond containing 10^6 gal of water is initially free of a certain undesirable chemical. Water containing 0.01 g/gal of chemicals flows into the pond at a rate of 300 g/h, and water also flows out of the pond at the same rate. Assume that the chemical is uniformly distribute throughout the pond. Let Q(t) be the amount of chemical in the pond at any time t. Write down the initial value problem for t. Solve the problem in part (a) for Q(t). How much chemical is in the pond after 1 year? After 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.
Explanation / Answer
2. a) Q(t) is the amount of chemical in the pond at any time t.
dQ/dt = 0.01* 300 - (Q/ 106) * 300
= 3 - 3Q/ 104
[ The first part is the amount of chemical going in and the second part is the amount going out]
b) The differential equation written above is a first order autonomous differential equation. The only solution to this is Q = 104
When Q < 104 , dQ/dt >0
When Q > 104 , dQ/dt <0
Therefore, Q = 104 is the only asymptotically stable solution.
Therefore, after one year Q = 104 and it is always the same as long as the chemical source is there.
c) After the end of one year, the chemical source is removed. Water rates flowing in and flowing out are the same.
The new differential equation is
dQ/dt = Q1y - (Q1y / 106) *300
where Q1y is the amount after one year (shown in part b).