Exploring Pollution in the Great Lakes
Goal: 2.05 Use recursively defined functions to model and solve problems
Equipment and materials needed for students:
1. Copy of handout, 1 Student Handout for each student.
2. Graphing calculator
3. Paper and pencil for note taking. Colored pencils.
Note: Your students should be familiar with
a. Writing recursive equations,
b. Using SEQUENCE mode on the calculator to generate numerical values, and
c. Using the calculator to graph the numerical values attained using sequence mode.
Most of the water flowing into Lake Erie comes from Lake Huron, and most of the water
flowing into Lake Ontario is from Lake Erie. Each year, 11% of the water in Lake Huron
flows into Lake Erie, while 36% of the water in Lake Erie flows into Lake Ontario, and
12% of the water in Lake Ontario flows out to the sea.
For generations, factories on the lakes had been dumping a pollutant into the water.
Presently, there are 4000 units of pollutant in Lake Huron, 2000 units in Lake Erie, and
3000 units in Lake Ontario. For the most part this form of pollution has stopped. Only
two such factories remain. One, on Lake Huron, is dumping 25 units of pollutant into the
water each year; the other on Lake Ontario is dumping 20 units of the pollutant into the
water each year.
How long will it be before the amount of pollutant in the three lakes is reduced to 10% of
its present level? What is the long-term level of pollutant in the lakes?
Source: Intermath: Four Sample Problems, COMAP, Inc., Lexington, MA, 1992.
TEACHER NOTES:
Let H n , En , and On represent the level of pollutants in year n for Lakes Huron, Erie and
Ontario, respectively. We can set up a set of recursive equations as follows:
AFM
Fall 2003
H n  H n 1  0.11H n 1  25
En  En 1  0.36 En 1  0.11H n 1
On  On 1  0.12On 1  0.36 En 1  20
With the original pollutant levels: H 0  4000 , E0  2000 , and O0  3000 .
Combining like terms in the recursive equations, we have:
H n  0.89 H n 1  25
En  0.64 En 1  0.11H n 1
On  0.88On 1  0.36 En 1  20
Using the Table feature, we see the value of the iterates below, where un , and vn
represent H n , and En , respectively. Unfortunately we are not able to see all three lists at
once. You can arrow over to see the values for wn .
The graph below shows the level of pollutants in each lake:
Window is Xmin = 0, Xmax = 30, Xscl = 5, Ymin = 0, Ymax = 4500, Yscl = 500
Where the level of pollutants in Lake Huron is represented by the solid trace type, Lake
Erie is the thick solid trace type and Lake Ontario is the dotted trace type.
AFM
Fall 2003
To answer the question “How long will it be before the amount of pollutant in the three
lakes is reduced to 10% of its present level?”, we will find what 10% of the original
pollutant levels are for each lake. These values are: 400, 200 and 300 units for Lakes
Huron, Erie and Ontario, respectively. Now we can trace on the graph above to find
when these levels are reached.
For Lake Huron, we see that the levels of pollutant is approximately 410 units after 26
years and 390 units after 27 years, so the level of pollutants is 400 units between 26 and
27 years. For Lake Erie, we see that the level of pollutants is approximately 219 units
after 21 years and 197 units after 22 years, so the level of pollutants is 200 units between
21 and 22 years. For Lake Ontario, we see that the level of pollutants is approximately
961 units when n is 30, so we need a bigger X window. Doubling the X window to 60
and using the trace key, we see that the level of pollutants is 403 after 60 years.
Experimenting with bigger windows, we should see that the level of pollutants in Lake
Ontario seems to level out to 375 units; therefore, the level of pollutants in Lake Ontario
will never reach 300 units.
In fact all of the pollutant levels seem to level off to an equilibrium value. For Lake
Huron, we notice that the equilibrium value is 227.27 units and for Lake Erie, the
equilibrium is 69.4 units. We have just answered the second question posed: “What is
the long-term level of pollutant in the lakes?”
We can find these values analytically by thinking about what it means for the level of
pollutants to stabilize. The pollutant level in any of the lakes will stabilize when the
amount of pollutants flowing out of the lake is equal to the amount of pollutants flowing
into the lake. Let’s consider our original recursive equation below for Lake Huron:
H n  H n1  0.11H n1  25
The amount of pollutants flowing out of Lake Huron is equal to the amount of pollutants
flowing into Lake Huron when
0.11H n1  25 .
Solving for H n 1 , we have
25
H n 1 
0.11
or H n1  227.27 .
Another way to think about finding the equilibrium value is to find the value for which
the iterates do not change. That is, finding the level of pollutants for which H n  H n1 .
AFM
Fall 2003
In our recursive equation, we can set H n  H n1 , and solve for H n as follows:
H n  H n 1  0.11H n 1  25
H n  0.89 H n 1  25
H n  0.89 H n  25
0.11H n  25
Hn 
25
0.11
Now we can use this equilibrium value to find the equilibrium values for Lakes Erie and
Ontario using analytic methods.
We will set En  En 1 and H n  H n1 .
En  0.64 En  0.11H n
0.36 En  0.11H n
En 
0.11
Hn
0.36
In the long run, we know H n  227.27 . Substituting this value in for H n , we have
En  69.4 . Similarly we can find the equilibrium value for the pollutant level in Lake
Ontario as shown below:
On  0.88On  0.36 En  20
0.12On  0.36 En  20
On 
0.36 En  20
0.12
Substituting the equilibrium value for the pollutant level in Lake Erie, we get On  375 .
This problem offers an interesting look at these dependent recursive equations and allows
us to solve for the equilibrium values analytically. The students have an opportunity to
consider numerical solutions and graphical solutions using the calculator and then
confirm those solutions using analytical methods.
AFM
Fall 2003
Exploring Pollution in the Great Lakes
Student Handout
Most of the water flowing into Lake Erie comes from Lake Huron, and most of the water flowing
into Lake Ontario is from Lake Erie. Each year, 11% of the water in Lake Huron flows into
Lake Erie, while 36% of the water in Lake Erie flows into Lake Ontario, and 12% of the
water in Lake Ontario flows out to the sea.
Figure 1: Map of the Great Lakes
For generations, factories on the lakes had been dumping a pollutant into the water. Presently,
there are 4000 units of pollutant in Lake Huron, 2000 units in Lake Erie, and 3000 units in Lake
Ontario. For the most part this form of pollution has stopped. Only two such factories remain.
One, on Lake Huron, is dumping 25 units of pollutant into the water each year; the other on Lake
Ontario is dumping 20 units of the pollutant into the water each year.
How long will it be before the amount of pollutant in the three lakes is reduced to 10% of its
present level? What is the long-term level of pollutant in the lakes?
Source: Intermath: Four Sample Problems, COMAP, Inc., Lexington, MA, 1992.
To help you answer the questions above we will explore the following:
a)
Write recursive equations to represent the amount of pollutants in each of the great
lakes. Include the initial values.
b)
Compare the levels of pollutant in the three lakes for the next 5 years. Give an
explanation for the increase or decrease in the level of pollution in each lake.
c)
Sketch a graph of the pollutant levels in each lake over time. Use colored pencils to
distinguish between the graphs.
d)
How many years will it take for the levels of pollutant in each of the three lakes to be
reduced to ten percent of each of their present levels?
e)
What is the long-term level of pollutant in the lakes. We can refer to these long-term
levels as equilibrium values. Could you predict the equilibrium value for Lake Huron using only
the recursive equation?
AFM
Fall 2003