Recursion Formulas and Limits to Water Quality
Teacher's Guide

Subject Area

This activity has been designed for the MAT 3A1, MCA 0A1, MFN 0A1 Mathematics curriculums. Specifically, Recursion Formulas and Limits to Water Pollution fits into MAT 3A1 after the section on Sequences, MCA 0A1 after the section on Limits, MFN 0A1 as an extension of the section on Markov Chains. This activity functions well as an opportunity to reinforce and apply learned skills.

Learning Outcomes

Teaching, learning and evaluation will focus on the student's ability to:

• Practice number skills, operations with decimals, fractions, and percentages;
• Review and consolidate skills in making and manipulating recursion formulas, and finding limits;
• Discover of the effect of chemical spills on water resources;
• Discover of the time required to purify contaminated water;
• Apply mathematics to predict the ultimate environmental damage caused by a constant source of pollution.

Classroom Development

1. Read the examples carefully and then gather the materials required to have the class complete the simulation lab. It is suggested that the teacher experiment with various dyes to find one that is suitable.
2. Before doing the lab, lead a class discussion drawing on existing student knowledge. Have students suggest and support their ideas as to how polluted the water will become.
3. Direct students to work cooperatively in small groups to complete the activity and answer the questions.
4. Have the class complete the lab using the student sheets provided.
5. Take up the answers as a class to ensure that the mathematics has been done correctly and to review the skills being used. Discuss the limits of the recursion process.
6. Ideally, some time will be set aside to allow students to discuss some of the implications of water pollution, chemical spills, oil spills, and water purification systems. Consider the position put forth by industry suggesting that as long as some cleaning is done, the environmental damage will be limited. Ask students if this is an acceptable excuse for environmental negligence.

Timing

Allow two periods for the completion of this activity.

Resource

The Mathematics Teacher. Drugs and Pollution in the Algebra Class. National Council of Teachers of Mathematics: February, 1992. (Vol. 85, No. 2.) (pp 139 - 145.).

Scenario

The volume of water in a lake is changed totally every 4 weeks. There is a chemical plant that is discharging soluble toxins into the lake once a week. How toxic will the lake become?

Simulation Lab

The lake in this scenario is a 4 L windshield washer fluid container, containing 2 L of water. Every "year" 500 ml of the water will be replaced.

The toxic spill is 16 ml of dye solution (mixture of food colouring and water).

Step 1: Remove 16 ml of water from the "lake."

Step 2: Pour 1/4 of the volume of the "lake" water into a beaker. Save this for future reference.

Step 3: Pour an equal volume of clean water into the lake.

Step 4: Pour 500 ml of clean water into the "lake."

Step 5: Repeat steps 1, 2, and 3 until an equilibrium is reached.

The Mathematics

If Tn is the amount of toxin in the "lake" after n time periods, then

Tn = 0.75 Tn-1 + 16

The first six terms of this sequence are: 16, 28, 37, 43.8, 52.6, 55.4

Example 1

How much toxin will be there after 10 weeks?

Solution

T7 = 0.75 T6 + 16 = 0.75 (55.4) + 16 = 57.5

T8 = 0.75 T7 + 16 = 0.75 (57.5) + 16 = 59.2

T9 = 0.75 T8 + 16 = 0.75 (59.2) + 16 = 60.3

T10 = 0.75 T9 + 16 = 0.75 (60.3) + 16 = 61.2

Therefore, after 10 weeks there will be 61.2 ml of toxin in the "lake."

Note: By continuing the calculations, it can be shown that after 25 weeks the amount of toxin in the "lake" is 63.96 ml. These calculations are very tedious! A better way would be to write a short computer program. The calculations indicate that the amount of toxin in the "lake" is approaching a limit.

This is how the limit can be calculated:

Let the limit be x ml of toxin.

(The amount of toxin remains constant for two successive weeks.)

Tn	=	Tn-1 = x
Tn	=	0.75 Tn-1 + 16
x	=	0.75x + 16
0.25x	=	16
x	=	64 mL

Therefore the amount of toxin in the "lake" reaches a limit of 64 ml.

Example 2

If a safe level of contaminants in the "lake" is 8 ml, what is the largest amount of toxin that the chemical plant can discharge weekly?

Solution

The limit of toxins is 8 ml.

Tn = Tn-1 = 8

Let the amount of weekly discharge be D ml.

Tn = 0.75Tn-1 + D

8 = 0.75(8) + D

D = 2 ml

The plant can safely discharge 2 ml of toxins each week

Recursion Formulas and Limits to Water Quality
Student's Guide

1. A lake has a volume of 150 million cubic meters. Each year, one fifth of the water is replaced by clean water. A chemical spill deposits 50 000 cubic meters of soluble toxic waste into the lake each year.

   (a) How much of this toxin will be in the lake after 3 years?

   (b) What will be the maximum amount of toxin that accumulates in the lake?

    

2. A circular swimming pool has a diameter of 10 m and an average depth of 1.5 m.

   (a) Calculate the volume of water in the pool.

   (b) For the water to be safe for swimming, it must be chlorinated. The concentration of chlorine must be 0.05% by volume. Each day, 15% of the chlorine in the pool is lost through evaporation and other mechanisms.

   (i) How much chlorine should be added to the pool initially to make it safe? This initial treatment is called a shock treatment.

   (ii) How much chlorine should be added to the pool each day to maintain the safety level?

   (iii) The pool can be made safe by adding the same amount of chlorine each day including the first day, avoiding the necessity of a shock treatment. Calculate the amount.

    

3. A water reservoir has a total volume of 150 000 cubic meters. The water in the reservoir is treated by an adjacent filtration plant. The plant can treat 18% of the total volume of water in the reservoir in one month. A local chemical plating company has been discharging effluent into the reservoir at the rate of 8000 L per day.

(a) How much of this effluent will be in the reservoir after six months?

(b) The water is safe to use so long as the concentration of chemical effluent is less than 2 ppm (parts per million). How many months did it take for the water become unsafe to use?
(c) If this pollution was allowed to continue unchecked, what would be the maximum amount of effluent that would accumulate in the reservoir?

(d) Calculate the concentration of chemical (from part c) in ppm.

(e) After 6 years, the pollution source was found and eliminated. How long will it take for the water to become safe to use again?