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Application
of Sequences to Water Pollution
Teacher's
Guide
Subject
Area
This activity
has been designed for the MAT 3A1, MTT 4G1, MAT 4A1, MFN 0A1 Mathematics
Curricula. Specifically, it fits into MAT 3A1 after the section
on Geometric Sequences, MTT 4G1 after the section on Exponential
Decay, MAT 4A1 after the section on Exponential and Logarithmic
Functions, MFN 0A1 after the section on Sequences.
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 exponential and logarithmic calculations;
- Become
aware of the effect of chemical spills on water resources;
- Become
aware of the time required to purify contaminated water;
- Learn
applications and implications of exponential decay.
Classroom
Development
- Teachers
should 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, have a class discussion about how long it should
take for the water to clear, as outlined in the student handout.
- Have
the class complete the lab. Display the beakers in a row to
illustrate the process.
- Students
could work cooperatively in small groups solving the questions.
- Use class
discussion time to see that the mathematics has been done correctly,
and to review the skills being used. Discuss the exponential
nature of decay.
- Spend
some class time to discuss some of the implications of water
pollution, chemical spills, oil spills, and water purification
systems.
Timing
Allow two
periods for the completion of this activity.
Resources
National
Council of Teachers of Mathematics. The Mathematics Teacher:
Drugs and Pollution in the Algebra Class. February, 1992. (Vol.
85, No. 2.) (pp 139 -145.).
Application
of Sequences to Water Pollution
Students's
guide
Scenario
The volume
of water in a lake is changed totally every 4 years. There is
a toxic chemical spill in the lake. How long will it take for
the lake to be clean again?
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 16 mL of "toxin" into the "lake."
- Step
3: Pour 500 mL of "lake" water into a beaker. Save
this water for future reference.
- Step
4: Pour 500 mL of clean water into the "lake."
- Step
5: Repeat steps 3 and 4 until the "lake" clears.
The Mathematics
If Tn is
the amount of toxin left after n time periods, then
Tn = 16
(0.75)n
The first
six terms of this sequence are:
16, 12,
9, 6.75, 5.06, 3.8
Example
1
How much
toxin will be left after 12 years?
Solution
T12 = 16
(0.75)12 = 0.5068 ml
Therefore,
after 12 years there will be 0.5068 ml of toxin left (3% of the
original spill).
Example
2
When will
75% of the toxin be eliminated?
Solution:
If 75% of
the toxin is eliminated then 25% is left.
| 25% of 16 ml |
= |
4 ml |
| 4 |
= |
16 (0.75)n |
| 0.25 |
= |
(0.75)n |
| log 0.25 |
= |
n log 0.75 |
| n |
= |
log 0.25/log 0.75 |
| n |
= |
4.8 years |
Alternate
Solution
(not using
logarithms)
0.25 = (0.75)n
Use your
calculator to estimate the value of n.
| n |
(0.75)n |
| 5 |
0.237 |
| 4 |
0.136 |
| 4.5 |
0.274 |
| 4.7 |
0.258 |
| 4.8 |
0.251 |
n = 4.8
75% of the
toxin will be eliminated after 4.8 years
- A lake
has a volume of 150 million cubic metres. Each year, one fifth
of the water is replaced by clean water. A chemical spill deposits
50 000 cubic metres of soluble toxic waste into the lake.
- (a)
How much of this toxin will be left in the lake after 3
years?
- (b)
How long will it take for the amount of toxic chemical in
the lake to be reduced to 5000 cubic metres?
- A chemical
spilled into a reservoir of pure water. The concentration of
the chemical in the contaminated water is 5%. In one month,
20% of the volume of the reservoir is replaced with clean water.
- (a)
What will the concentration of the contaminant be one year
from now?
- (b)
For water to be safe to drink, the concentration of this
chemical must be less than 0.005%. How long will it be before
this water is drinkable?
- A water
purification plant uses two filtration devices which operate
in sequence. The first device removes 20% of the contaminants
that pass through it, and the second device removes 30%. Water
is circulated continuously through these devices until 99% of
all contaminants are removed. How many times does the water
go through these devices?
- A small
swimming pond is approximately circular with a diameter of 500
m and an average depth of 4 m.
- (a)
Calculate the volume of the pond.
- (b)
On June 18th, a leak in a sewage line caused 10 000 L of
toxic waste to be dumped into the pond. Calculate the concentration
of toxins in the pond in parts per million (ppm). Note:
1 cubic metre = 1000 litres.
(c) A portable waste treatment plant was established on
the site to remove the toxin. The plant purifies 105 L of
the pond's water each week.
- (i)
What is the concentration of toxin in ppm after 6 weeks?
- (ii)
The pond concentration is considered safe for swimming
when the concentration of toxin is 4 ppm or less. On
what date will the pond be safe to swim in again?
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