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
- 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.
- 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.
- Direct
students to work cooperatively in small groups to complete the
activity and answer the questions.
- Have
the class complete the lab using the student sheets provided.
- 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.
- 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
- 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?
-
- 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.
-
- 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?
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