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.