# Crossroads in Mathematics

## Example 9

### Maintaining Pool Chlorine Level (Numerical Experimentation)

This problem presents students with a genuine problem situation for which they must make and
test conjectures about its solution. With the support of technology, students can experiment to
determine a solution to the problem.

**Problem.** Chlorine is used to control microorganisms in the water of a pool. Too much chlorine
produces burning eyes; too little, and slime develops. Here are some facts about pool care:

- Chlorine dissipates in reaction to bacteria and to the sun at a rate of about 15% of the amount
present per day.
- The optimal concentration of chlorine in a pool is from 1 to 2 parts per million (ppm), although
it is safe to swim when the concentration is as high as 3 ppm.
- It is normal practice to add small amounts of chlorine every day to maintain a concentration
within the 1 to 2 ppm ideal.

a. Use a calculator to find the amount of chlorine (in ppm) remaining each day for 10 days, if the
level at time zero is 3 ppm and no more is added.

b. Graph the concentration of chlorine (in ppm), *c*, as a function of time, *t*, for the data
determined in part a. Find the interval of time over which the chlorine level is optimal for humans.

c. If chlorine is added every day, another model is necessary. Use a computer or calculator
spreadsheet to model this system for a 21-day period when the concentration is 3 ppm at time
zero.

I. Try adding 1 ppm each day. Clearly that is too much, but does the pool water turn to
chlorine? What is the largest amount of chlorine attainable?

ii. Try adding 0.1 ppm everyday. Does this process yield ideal conditions in the long run?

iii. Find a daily dosage that stabilizes the concentration of chlorine at 1.5 ppm.

In solving this problem, students must understand thoroughly the parameters of the situation--the
facts about pool care. Through experimentation afforded by technology, students can design and
determine an effective solution to the chlorine problem and judge whether their solution is
reasonable.