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How’s the water?

Modeling with Exponential Functions

Module 24.1

Prepared for SSAC byVauhn Foster-Grahler – Evergreen State College

© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2005

Quantitative skills and conceptsData AnalysisMathematical ModelingLogarithmic Re-expressionSolving Logarithmic EquationsReading and interpreting graphs

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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.

Preview

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•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.

Overview of Module

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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!

Conceptual thinking

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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.

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.

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

1. Enter the data from this table into an Excel spreadsheet.

Concentration of algae-eating Bacteria (parts per million per hour)

0

5,000

10,000

15,000

0 2 4 6 8 10 12

Time (hrs)

Bac

teri

a C

once

ntra

tion

(ppm

)

7

Now we will use a Trendline to find a function that that describes the trend of the data.

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.”

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 R-squared value.

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?

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.

Procedure: Adding a Trendline

Concentration of algae-eating Bacteria (parts per million per hour)

y = 849.76x - 2995

R2 = 0.578

-5,000

0

5,000

10,000

15,000

0 2 4 6 8 10 12

Time (hrs)

Ba

cte

ria

C

on

ce

ntr

ati

on

(p

pm

)

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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”.

Procedure: Linearizing Data

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Create a new scatter plot with the re-expressed 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?

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.

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?

How does taking the logarithm of a number change the number?

Procedure: Linearizing Data

Change in ln(Concentration(ppm)) Over Time

0.000

2.000

4.000

6.000

8.000

10.000

0 2 4 6 8 10 12

Time (hrs)

ln(C

on

ce

ntr

ati

on

(pp

m))

Refer to slide 6 for hints on creating a scatter plot.

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Compare the two scatter plots.

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.

9. Do both of these graphs say the same thing?

Procedure: Analysis

Change in ln(Concentration(ppm)) Over Time

0.000

2.000

4.000

6.000

8.000

10.000

0 2 4 6 8 10 12

Time (hrs)

ln(C

once

ntra

tion(

ppm

))

Concentration of algae-eating Bacteria (parts per million per hour)

02,0004,0006,0008,000

10,00012,00014,000

0 2 4 6 8 10 12

Time (hrs)

Ba

cte

ria

C

on

ce

ntr

ati

on

(p

pm

)

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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.

This trendline, with an R2 value of 0.9887, more accurately describes the trend of the data.

Recall that the trendline of the raw data (slide 7), had an R2 value of 0.578.

Procedure: Analysis

Change in ln(Concentration(ppm)) Over Time

y = 0.6928x + 1.717

R2 = 0.9887

0.000

2.000

4.000

6.000

8.000

10.000

0 2 4 6 8 10 12

Time (hrs)

ln(C

on

ce

ntr

ati

on

(pp

m))

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.

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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?

Procedure: Analysis

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?

Which function is the best model?The next slide asks you to test one possible way of

determining which function is the best model.

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.

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“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.

Procedure: Determining which Function is the Best Model

Concentration of algae-eating Bacteria (parts per million per hour)

y = 5.568e0.6928x

R2 = 0.9887

-4,000-2,000

02,0004,0006,0008,000

10,00012,00014,000

0 2 4 6 8 10 12

Bacteria Concentration (ppm)

Tim

e (h

rs)

*

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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.