Expert: Jack Cheng - 11/17/2006

Question
Consider a pond that has an initial volume of 10000 cubic meters.
Suppose that at time t=0, the water in the pond is clean and that the pond
has two streams flowing into it, stream A and stream B, and one stream
flowing out, stream C. Suppose 500 cubic meters per day of water flow
into the pond from the stream A, 750 cubic meters from stream B, and
1250 cubic meters flow out of the pond via stream C.

At time t=0, the water flowing into the pond from stream A becomes
contaminated with road salt at a concentration of 5 kg per 1000 cubic
meters. Suppose the water is well mixed so that the concentration of salt at
any given time is constant. To make matters worse, suppose also that at
time t=0 people dump trash into the pond at a rate of 50 cubic meter
per day. The trash settles to the bottom of the pond, reducing the volumn
by 50 cubic meters per day. To adjust fo rthe incoming trash, the rate
that water flows out via stream C increases to 1300 cubic meters per
day and the banks of the pond do not overflow.


I need some help in writing an equation that I can solve for the amount
of salt in the water at time t.

Thank you,
Kathrine

Answer
Hi Kathrine,

I am assuming you want the concentration of the salt in the water, as to just amount of salt.

Let's decide that the unit for t is day.

First, you need to write an equation for the amount of salt entering the pond per day. The salt is only carried by Stream A, so the amount of salt that enters is related only to the concentration of salt in it and the 500 cubic meters per day that it supplies to the pond.

Next, you need to write an equation for the amount of water in the pond at day t. The amount is the initial amount plus the amounts flowing in (namely, that from Streams A and B) minus the amounts flowing out (namely, that from Stream C).  Then, you take into account the amount trash thrown into the pond, which also increases the volume of water. (I think the pond is going to overflow eventually).

Now, just divided the amount of salt by the amount of water, and you should have the concentration of salt in the pond.

I hope this is enough to get you started.

~ Jack