Rated MPG for Confusion by Bode, Gleue and Pearson

Part III – The Graph Explains All

What’s going on here? Cody’s parents had made what appeared to be the best decision. They traded in the Corolla to get the Hybrid, with an increase of 20 miles per gallon (mpg). However, the best choice seems to have been trading in the SUV for the Minivan with only a 4 mpg increase!

To really understand this conundrum, we have to look at the big picture. This means first looking at more data, graphing the data, and then obtaining the correct equation or function for this data. We looked at four cars and that was enough to answer a few simple questions. Looking at the big picture requires us to analyze more vehicles.

Table 4 lists 21 different vehicles with mpg ratings ranging from 9 to 48 mpg. Use this information to calculate the gallons of gas needed to drive the cars 100 miles.

Before you fill in the table, let’s have the computer do what it’s really good at—crunching numbers! See the directions after the table, as we are going to complete the rest of the activity on the computers.

Table 4

Car type and number	Distance (miles)	Miles per gallon (mpg)	Gallons of gas needed to go 100 miles
1. Lamborghini Murcielago	100	9
2. Chevy Avalanche 4WD SUV	100	11
3. Cadillac Escalade SUV	100	12
4. Dodge Viper Convertible	100	13
5. Hummer H3 4WD SUV	100	14
6. Chevy Corvette	100	15
7. Porsche 911	100	16
8. Toyota Sienna Minivan	100	17
9. Mitshubishi Eclipse Spyder	100	19
10. VW Beetle	100	20
11. Dodge Avenger	100	21
12. Honda Accord	100	22
13. Nissan Altima	100	23
14. Ford Focus	100	24
15. Hyundai Elantra	100	25
16. Minicooper	100	26
17. Toyota Corolla	100	28
18. Toyota Yaris	100	29
19. Ford Escape Hybrid SUV	100	34
20. Honda Civic Hybrid	100	40
21. Toyota Prius	100	48

Questions

1. Use a graphing program (such as Logger Pro 3 or Microsoft Excel) to calculate the gallons of gas needed to drive 100 miles (column 4 in Table 4 above). (For your convenience, a comma delimited version of Table 4, gas_mileage_data.txt, is available which can be imported into your program. For example, in Excel, go to Data→Import External Data→Import Data..., navigate to where you downloaded the file, and then walk through the wizard to identify the data as delimited, using commas as delimiters.)

2. Now we are ready to graph the data. For this graph, plot “miles per gallon” as the x variable and “gallons to go 100 miles” as the y variable. Print your graph.

3. Is your graph a straight line?

4. Describe the appearance of your graph. As one variable gets larger, what happens to the other variable?

5. Offer a suggestion for the type of graph (or function) shown.

6. Use your graphing program to determine the type of function or graph that best fits this data. At this point, do you need to modify your answer given in Question 5 above, yes or no? If so, modify it now.

7. Use your graphing program to generate the equation for this plot. The curve generated will follow the points as best as it can. What is the equation that describes your graph?

8. Your equation will have a constant of proportionality for this function. The constant of proportionality is often symbolized as either A or k.

   1. What is your constant of proportionality?

   2. We have seen this number before. Describe in words what this number means.

9. Your graphing program uses general variables, x and y. However, in our example, x and y stand for specific variables!

   1. What was our x variable (see horizontal label on the graph)?

   2. What was our y variable (see vertical label on the graph)?

   3. Now, rewrite the equation you wrote in Question 9 to match your set of variables.

   4. Verbalize the meaning of this graph in a sentence.

10. Now, use your graphing program to calculate the cost to drive 100 miles in each vehicle in Table 4. Label the new column “cost to drive 100 miles,” and assume that gas costs $4 per gallon. The units will be “dollars.”

11. Plot this new data with “miles per gallon” as the x variable and “cost to drive 100 miles” as the y variable.

12. Use your graphing program to determine the type of function or graph that best fits this data.

    1. What type (or function) is this graph?

    2. Write the equation using our variables.

13. What is the constant of proportionality for this graph?

14. What is the significance of this number (or how did the software come up with this number)?

15. Now let’s go back to the original issue at hand. In Part I, you made a hypothesis for the choice of cars the Andrews family should trade in. There were two choices: Cody’s family could keep the SUV and trade the Corolla for the Hybrid OR they could trade the SUV for a Minivan and keep the Corolla.

    In Table 1 of Part II, you calculated the gallons of gas needed to drive all four cars 100 miles and the cost to go this distance. Use your graphs to verify these data points.

16. It was noted earlier that it would have been a better choice economically for Cody’s parents to trade in the 13 mpg SUV for the 17 mpg Minivan instead of trading in the 28 mpg Corolla for the 48 mpg Hybrid. Look at your “cost versus mpg” graph and offer an explanation for the reason for this.

17. Knowing the type of graph and its mathematical function helps to understand the relationship between cost and mpg rating. When asked to choose between trading in the 13 mpg SUV for the 17 mpg Minivan OR trading in the 28 mpg Corolla for the 48 mpg Hybrid, lots of people think the best decision economically would be to trade in the Corolla. All other variables being equal, this wouldn’t be the best choice. Go home and ask your parents what they think about it. Why do you suppose most people choose the wrong answer?

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