You are the manager of a swimming pool which contains 106 liters of water.
You added 5*105 mg of Cl to the pool, for a concentration of ˝ mg Cl/liter of water, just like your manager’s handbook says you should. But your assistant also added 5*105 mg Cl, and now the concentration of chlorine is twice what it should be. Rather than shutting everything down so that you could empty the pool and then refill it, you decide to drain the water and simultaneously replenish it via a hose containing fresh water. That way, you don’t need to send the swimmers home. (A chlorine concentration at this level may be irritating, but it is not so high as to be a danger to your patrons).
1. Suppose water is drained out at a rate of 2000 liters/min, and fresh water is added at the same rate. The pool water is kept thoroughly mixed, so the concentration of chlorine in the drained water is the same as it is overall in the pool
a. Set up a differential equation, together with initial conditions, which models the total amount of chlorine in the pool after t minutes.
Let y(t) be the number of mg of Cl in the water after t minutes. No new chlorine is being added, but since 2000 liters of water out of the 106 are being removed each minute, 2000/106 of the chlorine is removed each minute too, so
b. Solve the initial value problem.
c. How long will it take until you should close the drain valve and turn off the hose?
2. Oops! We forgot a very important consideration. The water coming out of the hose has a small amount of chlorine in it, too. Suppose it contains 0.1 mg/liter. Repeat all 3 parts of question 1.
Now, when you change the water, chlorine is both added and removed. It is added at a rate of
, and removed at the same relative rate as before. So
Given the initial condition, we now solve for C=900,000, so
We should close the drain valve and turn off the hose when y=500,000, so
Also – please notice (if you have not already done so) that this version of the problem has an equilibrium at y=100,000. What is the physical significance of this? Why is this “obvious”?