You are the manager of a
swimming pool which contains 10^{6} liters of water.

You added 5*10^{5 }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*10^{5 }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 10 ^{6} are being removed each minute,
2000/10^{6} 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”?*