Algebra 2 Spring Project
Mathematical Modeling

Due date: Mon, March 21 by 3:45pm
Introduction of Project Fri, Feb 26
Work on Project Thu, Mar 3 (7th)
Tue, Mar 15 (5th)
Work on Project Mon, Mar 14
Work on Project Mon, Mar 21
PROJECT DUE BY 3:45pm
(regardless of your attendance that day)
Project is due for LATE PENALTY Wed, Mar 23 by 8pm

Over the course of this year, you have learned about various functions (linear, quadratic, higher degree polynomials, exponential, and logarithmic). This project will allow you to inspect a set of real-world data, look at its graph, determined the kind of model that fits it best, use your calculator to find this model, and answer some questions using the equation.

For each of the three sets of data, produce a small report containing the following:
• A description of the problem; a table with the original data
• All graphs must be very neat graph on graph paper (by hand or using technology), titled, labeled, and with a reasonable scale.
• A response to each of the questions (retype each question and give your complete response – always explain why you’re doing what you’re doing.).
• All mathematical working must be shown (typed), using the correct notation and reasonable rounding. Be sure to use Microsoft Equation Tools to help you with mathematical symbols such as fractions and square roots.

Please note:
• This will be graded for accuracy, completion, neatness, and following instructions. It will be worth one quiz grade. See rubric for grading details.
• All work must be done individually.
• No outside help allowed – from other students, teachers, tutors, etc. If you receive help from someone other than your math teacher, it will be considered an honor code violation.
• Please do NOT turn the project in with a plastic cover. Please do NOT make a coversheet.

Submitting the project electronically?
Students should save their file in the following format:
• “Modeling period last name, first name.docx”.
• For example, Susie Jones in 3rd period would use “Modeling 3 Jones, Susie.docx” as her filename.
You may turn in a paper copy of your project. I recommend you do this especially if you refer to the color of your graphs, in which case you will have to print on a color printer.
If you wish to turn your project in electronically, please turn it in on Haiku. Click on the Project on the calendar, then click on “Hand In.”
I. Stopping Distances

When a driver stops her car, she must first think to apply the brakes. Then the brakes must actually stop the vehicle.

The table below lists average times for these processes at various speeds.

Speed (km/h) Thinking distance (m) Braking distance (m)
32 6 6
48 8 14
64 12 24
80 16 38
96 17 55
112 22 75

In this task you will develop individual functions that model the relationships between speed and thinking distance, as well as speed and braking distance.
1. Usegraphing software to create a scatterplot of speed versus thinking distance. Be sure to label the axes and give your graph a title.What type of function could model the behavior of the graph? Explain why you chose this function.
2. Use your knowledge of functions to develop a function (using a few points from the data) that best represents speed versus thinking distance. Show your work. Explain your process. Round to the nearest thousandth, where necessary.
3. Graph your function with the scatterplot on a new set of axes. Comment on how well your function fits the data. Be sure to include the equation of the function in the graph.
4. Use technology to find a regression function that best fits speed versus thinking distance. Round to the nearest thousandth, where necessary.
5. Graph the regression function with the scatterplot on a new set of axes. Comment on how well the function fits the data. Be sure to include the equation of the function in the graph.
6. On a new set of axes, draw both of the functions. Comment on any similarities or differences.
7. REPEAT STEPS 1 TO 6 FOR SPEED VERSUS BRAKING DISTANCE.
8. Estimate the thinking distance for a speed of 70 kmph. Estimate the braking distance for a speed of 55 kmph. Show your work. Explain your process.
9. Predict the braking distance for a speed of 125 kmph. Show your work. Explain your process.
II. Does fast driving waste fuel?

How does the fuel economy of a car change as its speed increases? Here are data for a Ford Escort. Speed is measured in miles per hour, and fuel economy is measured in miles traveled per one gallon of gasoline.

Speed (mph) Fuel economy (mpg)
1 15 22.3
2 20 25.5
3 25 27.5
4 30 29.0
5 35 28.8
6 40 30.0
7 45 29.9
8 50 30.2
9 55 30.4
10 60 28.8
11 65 27.4
12 70 25.3
1. Use graphing software to create a scatterplot of the data. Be sure to label the axes and give your graph a title. What type of function could model the behavior of the graph? Explain why you chose this function.
2. Use your knowledge of functions to develop a function (using a few points from the data) that relates speed and fuel economy. Show your work. Explain your process. Round to the nearest thousandth, where necessary.
3. Graph your function with the scatterplot on a new set of axes. Comment on how well your function fits the data. Be sure to include the equation of the function in the graph.
4. Use technology to find a regression function that best fits the data. Round to the nearest thousandth, where necessary.
5. Graph the regression function with the scatterplot on a new set of axes. Comment on how well the function fits the data. Be sure to include the equation of the function in the graph.
6. On a new set of axes, draw both of the functions. Comment on any similarities or differences.
7. Calculate the most efficient speed for this particular car to be driven. Show your work. Explain your process. Is this feasible? Why or why not?
8. Use one of the functions to calculate the fuel economy in miles per gallon if you are traveling 63 mph? Show your work. Explain your process.
III. Cell Phone Problem

In 1990, there were 205 cell phone subscribers in the small town of Centerville. The number of cell phone subscribers has increased dramatically according to the chart below.

Year Number of cell phone subscribers
1 1990 205
2 1991 298
3 1992 870
4 1993 1930
5 1994 2792
6 1995 4478
7 1996 8076
8 1997 15335
9 1998 27040
10 1999 39082

1. Use graphing software to create a scatterplot of the number of cell phone subscribers versus the number of years since 1990. Be sure to label the axes and give your graph a title. What type of function could model the behavior of the graph? Explain why you chose this function.
2. Use your knowledge of functions to develop a function (using a few points from the data) that relates the number of cell phone subscribers and years since 1990. Show your work. Explain your process. Round to the nearest thousandth, where necessary.
3. Graph your function with the scatterplot on a new set of axes. Comment on how well your function fits the data. Be sure to include the equation of the function in the graph.
4. Use technology to find a regression function that best fits the data. Round to the nearest thousandth, where necessary.
5. Graph the regression function with the scatterplot on a new set of axes. Comment on how well the function fits the data. Be sure to include the equation of the function in the graph.
6. On a new set of axes, draw both of the functions. Comment on any similarities or differences.
7. Using the regression function, how many cell phone subscribers will there be in Centerville in the year when you turn 30 years old?Show your work. Explain your process. Round to the nearest thousandth, where necessary.