Download What is Mathematical Modeling? - SERC

Module 24.1
Modeling with Exponential Functions
How’s the water?
Prepared for SSAC by
Vauhn Foster-Grahler – Evergreen State College
Quantitative skills and concepts
Data Analysis
Mathematical Modeling
Logarithmic Re-expression
Solving Logarithmic Equations
Reading and interpreting graphs
© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2005
1
Preview
It is a few centuries into the future. The Terran population explosion and
the concurrent demise of natural resources has spurred a group of people to seek
out a new home in a far, far away galaxy.
The Terrans, who are quite adept at space travel are in orbit around the
planet Riker in the Picard system. While the planet’s atmospheric composition is
sufficient to sustain the Terrans, the water supply on the planet is tainted with an
invisible algae that is toxic to all carbon-based life forms (this includes Terrans).
Without thinking of the consequences of their actions, the Terrans have
developed a bacteria that feeds on a substance in the algae. In sufficient
concentration the bacteria, which is harmless to Terrans, prevents the algae from
growing. The bacteria behaves similarly to many invasive species on Earth –
English ivy, crab grass, buttercup, blackberries, etc.
The Terrans have collected data on the growth habits of the bacteria in the
lab aboard the spacecraft. They will use these data to predict when the water on
Riker will be safe and the Terrrans can begin colonization.
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Overview of Module
•Slide 3:
•Discusses mathematical modeling
•Slide 4:
•Identifies the problem
•Slide 5:
•Presents data about the growth rate of the bacteria
•Slides 6 and 7:
• Ask you to graph the data and find and graph the linear regression
equation that models the data.
•Slide 8:
•You will use logarithmic re-expression to Iinearize the data
•Slides 9 and 10:
•Examines the effects of taking the logarithm and asks you to compare
the original scatter plot with the linearized data
•Slides 11-13:
•Uses linear regression and algebraic techniques to find an exponential
function to model the data and evaluate the effectiveness of the model.
•Slide 14:
•The end-of-module assignment.
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Conceptual thinking
What is Mathematical Modeling?
Mathematical modeling takes many forms.
•It is what school districts use to determine in the spring how many
teachers they will need in the fall.
•It is how you calculate how much money you must save each month to
retire at 55 as a millionaire (or not have to work two jobs in the summer).
One type of mathematical modeling looks for ways to describe trends in
data using a least squares or linear regression line. Sometimes, data
that do not appear linear are “linearized” or non-linear regression lines
are used to model them.
In this module you will be asked to find a function to model data about
the growth of a mythical bacteria to determine when the water on a
planet will become potable and the planet habitable by Terrans. Enjoy!
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The Problem
•We know that when the concentration of algae-eating
bacteria reaches 250,000 parts per million the algae
can no longer survive and the water will be safe for
the Terrans to use.
•Since we don’t have a lot of time to collect data, we
want to find a function that models the change in
concentration of the bacteria.
•We will then be able to use this function to
extrapolate when the water will be safe to use and the
colonization of the planet Riker can begin.
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Procedure
2. The next task, is to use excel to create a scatter plot
of the change in concentration of the bacteria.
a)
Highlight the cells containing the data you want to graph. If the
data are not in adjacent columns highlight the left most column,
press the control key, release the left click and move the mouse
to the other column you want to include on your graph.
Highlight these cells the same way. Release the left click.
b)
Click on the chart wizard, select scatter plot and follow the
commands
3. Does your graph have a title and are the axes
labeled? (If not, use the chart wizard tabs to make these corrections.)
= Cell with a number in it.
= Cell with an equation in it.
1. Enter the data from this
table into an Excel
spreadsheet.
Bacteria
Concentration (ppm)
Concentration of algae-eating Bacteria
(parts per million per hour)
15,000
10,000
5,000
0
0
2
4
6
Time (hrs)
8
10
12
6
Procedure: Adding a Trendline
1. Move the cursor to one of the data
points on the graph, right click, select
“Add Trendline”.
2. Under the “Type” tab select “Linear.”
Concentration of algae-eating Bacteria (parts per
million per hour)
Bacteria
Concentration (ppm)
Now we will use a Trendline to find a
function that that describes the
trend of the data.
15,000
10,000
y = 849.76x - 2995
R2 = 0.578
5,000
0
0
2
4
6
8
10
12
-5,000
Time (hrs)
3. Under the “Options” tab turn on
“Display equation on chart” and
“Display R-squared value on chart.”
4.
Then click on “OK.”
5. Record the equation and the Rsquared value.
R2 is used to determine how well a function
describes the trend of the data.
The closer the R2 value is to 1, the better
the fit of the Trendline to the data.
1. Do you think the Trendline is a good fit
for the data? Why or why not?
2. Think about the basic function shapes
we’ve learned in class. Which function
does the scatter plot remind you of?
Explain.
3. Assuming you suggested an
exponential function. What function
could we apply to the dependent
variable to make our data more linear?
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Procedure: Linearizing Data
We can use the logarithmic function
(lnx) to linearize the data.
Use the column to the right of the last
column on your Excel spreadsheet to take
the natural logarithm of the dependent
variable, concentration of bacteria.
In cell c3 input the formula =ln(b3)
You can transfer this formula to the cells below
by moving the cursor to the bottom right corner
of the box that contains the formula and dragging
it down over the cells you want to contain the
formula.
To round to three decimal places, highlight the
cells you want to format, right click and scroll
down to “Format Cells”. Click on the “Number”
tab, highlight “Number” and indicate 3 decimal
places and click “OK”.
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Procedure: Linearizing Data
Create a new scatter plot with the reexpressed data (Column c) as the
dependent variable. The time will still be
the independent variable.
4.Do the data appear more linear?
Why or why not?
Refer to slide 6 for hints on
creating a scatter plot.
5.Examine the scale of the new
graph and compare it to the scale
of the original plot you created in
slide 6. Describe and explain any
differences between the two
scales.
ln(Concentration(ppm))
Change in ln(Concentration(ppm)) Over Time
10.000
8.000
6.000
4.000
2.000
0.000
0
2
4
6
8
10
Time (hrs)
How does taking the logarithm of a
number change the number?
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6. Why did Excel automatically
change the scale on the graph?
What would the graph on this slide
look like if you graphed it on the
scale from the graph on slide 6?
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Procedure: Analysis
Bacteria
Concentration (ppm)
Concentration of algae-eating Bacteria
(parts per million per hour)
14,000
12,000
10,000
8,000
6,000
4,000
2,000
0
Compare the two
scatter plots.
0
2
4
6
8
10
12
Time (hrs)
ln(Concentration(ppm))
Change in ln(Concentration(ppm)) Over Time
7. What do you think? Was our
attempt to linearize the data
by taking the natural
logarithm of the dependent
variable successful?
8. What are the implications of
linearizing the data.
10.000
8.000
6.000
9. Do both of these graphs say
the same thing?
4.000
2.000
0.000
0
2
4
6
8
10
12
Time (hrs)
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Procedure: Analysis
As you can see from the scatter plots on the previous slide taking the
logarithm of the dependent variable when data appear exponential linearizes
the data. Now let’s look at this more formally.
Investigate the linearity of the re-expressed data by finding a linear
trendline to fit the data.
Use the techniques on slide 7 to find the linear Trendline for the re-expressed
data. Remember to display your equation and R-Squared values.
ln(Concentration(ppm))
Change in ln(Concentration(ppm)) Over Time
This trendline, with an R2
value of 0.9887, more
accurately describes the
trend of the data.
10.000
y = 0.6928x + 1.717
8.000
2
R = 0.9887
6.000
4.000
Recall that the trendline of
the raw data (slide 7), had an
R2 value of 0.578.
2.000
0.000
0
2
4
6
Time (hrs)
8
10
12
11
Procedure: Analysis
BUT… We are trying to determine when the concentration of bacteria is sufficient for a safe
water supply and the previous graph is the natural logarithm of the concentration.
How can we get back to our original question of modeling the change in the concentration of
bacteria?
The linear regression equation for the graph on the previous slide
is (1) y = 0.6928x +1.717.
But because we took the natural logarithm of the dependent
variable the equation is really
(2) ln(y)= 0.6928x +1.717
Solving this equation for y, will result in an exponential function
that can be used to describe the original data.
So now we have two functions that model the concentration of
bacteria over time.
Which function is the best model?
The next slide asks you to test one possible way of
determining which function is the best model.
10. Solve equation (2) for y.
Show all your work.
Now we have two
functions to describe the
data. One function, the
linear regression
equation, and the second
an exponential function that
you calculated in question
1.
11. What are three ways that
we can compare the
outputs for these two
mathematical models with
the actual data?
12
Procedure: Determining which Function is the Best Model
“Which function is the best model for the data?”
One way we can compare the models is to graph each with the original scatter plot.
Use excel create a graph that displays the original scatter plot, the linear regression graph,
and the graph of the exponential function* you found in the previous slide.
*If you can’t figure
out how to input your
equation into Excel,
you can go to “Add
Trendline” and create
a exponential graph
to match your data.
Time (hrs)
Concentration of algae-eating Bacteria
(parts per million per hour)
14,000
12,000
10,000
8,000
6,000
4,000
2,000
0
-2,000 0
-4,000
y = 5.568e0.6928x
2
R = 0.9887
*
2
4
6
8
10
12
Bacteria Concentration (ppm)
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END OF MODULE ASSIGNMENTS
From the module:
1. Turn in or e-mail answers to the questions (#1-11) on slides 7, 9, 10, and
12. Use complete sentences.
2. Print out or e-mail your spreadsheets and graphs from slides 6-11, and 13.
And then- e-mail or turn in answers to the following questions…
1. As you compare the graphs on slide 13 with the original data, what are your
thoughts?
2. Which of the two functions would do a better job of modeling the data and
predicting when the concentration of bacteria will reach 250,000 parts per million?
3. When will the bacteria reach the required concentration? Show all your work and
use at least two methods to find your solution.
4. How confident are you that you have found a good model? In other words, would
you be the first to drink the water?
5. On slide 12 you were asked to come up with 3 ways to compare the two models
with the original data. Explain and present one of the ways (not including graphical
comparison) you suggested.
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