SECONDARY SCHOOL MATHEMATICS FROM THE EPA GAS MILEAGE GUIDE

SECONDARY SCHOOL MATHEMATICS FROM THE EPA GAS MILEAGE GUIDEAuthor(s): FLOYD VESTSource: The Mathematics Teacher, Vol. 72, No. 1 (JANUARY 1979), pp. 10-14Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27961500 .

Accessed: 13/09/2014 11:40

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 75.10.116.250 on Sat, 13 Sep 2014 11:40:19 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=nctm

http://www.jstor.org/stable/27961500?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


SECONDARY SCHOOL MATHEMATICS FROM THE EPA GAS MILEAGE GUIDE

Some timely applications of algebraic ideas and skills.

By FLOYD VEST North Texas State University

Dent?n, TX 76203

It is particularly nice when realistic appli cations of secondary school mathematics are found in areas where students have a

personal interest. Many Americans are in

terested in automobile fuel economy. As we

shall see in the following examples, practi cal decisions involving fuel economy re

quire important skills from secondary school algebra and analysis.

Calculating Average Gas Mileage

The gas mileage guide published by the Environmental Protection Agency (EPA) give a combined average for city and high way driving. Consider the following ex

ample from the 1977 EPA guide (pp. 16,

19), given in table 1.

TABLE 1 EPA Gas Mileage

Corn

High- bined

Engine City way Average

Pontiac Catalina

Buick

Century

231/V6

231/V6

17

16

25

26

20

19

To check this data, we compute the arith

metic average for the gas mileage for the

Pontiac:

17 + 25 = 21

This is not the same as the EPA average. We find a more dramatic difference when we check the average for the Buick Cen

tury:

EPA's figure is 19. This is quite a discrep ancy. How does EPA compute these aver

ages? Their average is based on the assump tion that half the driving is done in the city and half on the highway. Consider, for ex

ample, the following case of the Pontiac

and the assumptions on which the mileage calculations are based:

100 miles driven at 17 mpg in the city

100 miles driven at 25 mpg on the highway

This comes to a total distance of 200 miles

and a total consumption of 100/17 plus 100/25 gallons of gasoline. Gas mileage is

therefore 200 2

100 100 17 25

= 2(17)(25) _ 17 + 25

17 ^

25

20.24 mpg.

This figure is much closer to the EPA com bined average for the Pontiac than is the arithmetic mean. This EPA average is our

old friend, the harmonic mean. For other uses of the harmonic mean, see Skidell

(1977). A few questions for students: Does the

arbitary but convenient choice of 100 miles affect the outcome of the computation?

What is a general formula for EPA's

weighted combined average of mpg in the

city and y mpg on the highway?

Improved Gas Mileage?Cost Impact

We next consider an important question about fuel economy. If one improves gas

mileage from 10 mpg to 15 mpg, would the annual savings in fuel cost be the same as

that from improving gas mileage from 15

mpg to 20 mpg? What would be the re

sponse of the average person or student to

10 Mathematics Teacher



this question? Let us examine the question. As in the 1977 EPA guide, we assume an

average of 60? per gallon for gasoline and 10 000 miles driven per year. The savings from 10 mpg to 15 mpg would be

10 000(.6Q) 10 000(.60) 10 15

= 10 000(.60) 15 - 10

(10X15) = $200

The savings from 15 mpg to 20 mpg would be

10 000(.60) 20- 15

(20X15) = $100

These results are quite surprising to some

people. Why such a difference in savings? In order to investigate this question, let us consider the following problem: If a car is

getting mpg, how much savings results from improving mileage to (x + 1) mpg?

Let y be dollars saved with gasoline cost

ing 60? per gallon and 10 000 miles being driven per year. Gas mileage is improved from mpg to + 1 mpg.

= 10 000(.6Q) 10 000(.60) + 1

10 000(.60) 1

x(x + 1) We shall refer to this function as the

"savings function." To the mathematics

teacher, it is an interesting function and can be examined for asymptotes, minima, and so on. What does it tell us about savings from improved gas mileage?

Consider the graph of the savings func tion for > 0 (see fig. 1, table 2). This graph

TABLE 2 Approximate Values for Graph in Figure 1

(mpg) y (savings)

0 10 15 20 25 30

Undefined 55 25 14 9 6

illustrates dramatically the phenomenon of the decrease in savings as gas mileage is

iiilij

?. ?.. . ."5 '- ,/ .. ?. ..??TOi

Fig. 1. Graph of savings function

improved. The graph drops more sharply between 10 mpg and 15 mpg than for

regions of the same width to the right. Many automobile users would consider the

savings of $300 from increasing fuel econ

omy from 10 mpg to 20 mpg to be quite substantial, but they would find the addi tional savings of

10 000(.60) 30-20

(30)(20) = $100

from increasing fuel economy from 20 mpg to 30 mpg not worth the personal inconven ience required in achieving the additional fuel economy. Perhaps U.S. automobile manufacturers are thinking of such a curve

(fig. 1) when they produce large cars, mid sized cars, and compact cars, many of which have EPA combined highway-city averages of 18 to 21 mpg. For many car

buyers, the decision to choose a car averag ing 20 mpg over one averaging 25 mpg reduces to (see fig. 1)

14 + 9 = $11.50,

the average savings per one mpg per year, which may not be worth it.

By way of discussing other rational func

tions, the teacher could ask, What kind of function would arise from increasing the

gas mileage from - 1 to mpg, from to + 5 mpg, from - 5 mpg to mpg?

January 1979 11



Impact of Price Increase

What would be the effect of a dramatic increase in gasoline prices? More particu larly, if the price of gasoline increases 20

percent, what happens to the savings func tion? (The reader can determine the answer to this question by examining the equation of the savings function.) With such an in crease in price, would a substantial change occur in our judgment of which gasoline mileage is reasonably economical?

To investigate these questions, we ask how the fuel cost per 10 000 miles increases as the price of gasoline increases 10? a gal lon for gasoline costing dollars per gallon and a car getting mpg. Does such an increase create the same increase in annual fuel costs at all ranges of fuel cost per gal lon? We present the following derivation,

which answers some of these questions:

Let be the cost per gallon of gasoline in

dollars; 4- .10 is 10? greater. Let be the

gasoline mileage in mpg. Let y be the

change in fuel cost arising from an increase of \0fi for 10 000 miles of driving.

= 10 000(z + .10) lOOOOz _ 1000 ^ XX

The graph for > 0 is shown in figure 2. y

Fig. 2. Graph of y =

is the change in dollars of fuel cost per 10 000 miles for a 10? increase in the price of a gallon of gasoline, is the gas mileage in mpg.

For the question about the effect of a 10? increase in the price of gasoline on the an

nual cost of fuel, the graph indicates that a driver getting 10 mpg would have an in crease of $100, a driver averaging 20 mpg would have an increase of $50, and one

averaging 30 mpg would have an increase of about $33 in fuel cost. Also, in our calcu lations with the immediately preceding equation, subtracted out, revealing that

The EPA average is not the arithmetic mean.

the increase of 10? per gallon would create the same increase in annual fuel cost at all

ranges of price per gallon. Additional questions one might ask of

students are these: How does the fuel cost

per 10 000 miles change for an increase of

20? per gallon? What increase in gas mile

age would compensate for this increase in fuel costs? Is there an equation for this?

What does the savings function look like if it is expressed as a function of two vari

ables, one of them being fuel costs per gal lon?

Lead-Free or Regular Gasoline? Cost Considerations

Some vehicles use regular gasoline, which often costs H a gallon less than lead free gasoline. This fact suggests certain

questions: How much more does it cost to use lead-free gasoline? Does it depend on the gas mileage or the cost per gallon of fuel? What is an equation for this variable and what is its graph? An increase of fuel cost of 3? per gallon is offset by how much improvement in fuel economy?

For this last question consider the fol

lowing assumptions and derivation: As sume lead-free gasoline costs 60? per gallon and regular costs 57? and total miles driven is 10 000. Let be the base fuel economy in

mpg. Let y be the increase in fuel economy in mpg above that offsets an increase in

price from 57? to 60?




!OOog(.57) = loooo

v ' + yy '

.Six -h .Sly = .60*

1

\9J

We have found a simple equation of a line with an interesting meaning for slope.

At a base gas mileage of 19 mpg, an in crease in economy of 1 mpg offsets the in crease in fuel costs. At a base of 28 mpg an increase of 2 mpg provides the offset, and at 9.5 mpg an increase of 0.5 mpg makes up for the increase from Sii to 60? per gallon. Vehicles burning lead-free gasoline may of ten get this much better gas mileage than

Savings are smaller when going from 20 mpg to 30 mpg than from 10 mpg to 20

mpg._

those burning regular gasoline. The savings many purchasers expected from regular gasoline may not have been achieved. The

savings should also be evaluated in terms of environmental concerns as well as possible future problems related to the availability of leaded gasoline. We could ask some ad ditional questions: What happens to the

slope of this function as the price of gaso line increases but the difference between the

prices of lead-free and regular remains the same? What does this mean in comparing vehicles with lead-free and regular gasoline in an era of higher gasoline prices?

Impact of Tune-up

According to EPA, "On the average a

tuned-up vehicle gets approximately 3 to 9

percent better fuel economy than one that has not been properly maintained" (EPA 1977, p. 7). Assuming a 6 percent better

economy from a tune-up, does one save

enough in fuel cost in 10 000 miles'to pay for a $30 tune-up? What factors does this

depend on? What does "6 percent better

economy" mean? (We have derived an an swer: For 10 000 miles, 15 mpg, an increase of .06(15) mpg, and gasoline costing 60? a

gallon, one saves $22.64. The reader may wish to check this answer.) What happens to such savings when gasoline prices in

crease, when miles driven increase, or when

mpg increase? What is an equation ex

pressing savings as a function of one of these variables? Can we get a tune-up for $22.64? How much does it cost to tune your own car? Do radial tires, which are claimed to save 5 percent on fuel, save enough in fuel to make up the additional cost? What mathematical concepts appear in such in

vestigations?

Impact of Driving Speed

It is well known that fast driving reduces fuel economy. The EPA manual says that fuel efficiency is "about 10 to 15 percent less for every 10 mph above 50 mph" (EPA 1977, p. 8). What does this mean? Calculus students and some analysis students would be interested in the following customary scientific interpretation and calculus deri vation.

Let y be fuel economy in mpg and s be the number of 10 mph over 50 mph. Thus s = 0 at 50 mph and s = 1 at 60 mph. Assume a reduction of 15 percent in fuel economy for each 10 mph above 50 mph.

ds = -

.157

-ASds +\nk

In y = ?ASs + In k

y = ke~A5s

Assume y = 15 mpg when s = 0 (50 mph). Thus we have the exponential equation

y = lSe~15s

Having a little fun with our calculator, we

generate the data in table 3.

TABLE 3 Impact of Driving Speed on Economy

^(mpg) mph

15 12.91 11.11 9.56 8.23 7.09

50 60 70 80 90 100

January 1979 13



The reader might try a factor less than 15

percent in this derivation to see if it gener ates more reasonable data. Is the EPA in formation in accordance with the reader's

experience?

Savings from Improved Gas

Mileage?a More General Problem

More advanced or enthusiastic students might enjoy investigating a dependent vari able, which is a function of two independ ent variables, by use of a table or a three dimensional model of a surface. Table 4

TABLE 4 Savings from Improved Economy

Average Fuel Costs per 15 000 Miles $812 $750 $696 $650 $609

Combined

mpg_12 13 14 15 16 12 0 62 116 162 203 13 -62 0 54 100 141 14 -116 -54 0 46 87 15 -162 -100 -46 0 41 16 -203 -141 -87 -41 0

summarizes data from the 1977 EPA guide and represents such a surface in two octants of a three-dimensional system. This table is based on the assumption that

gasoline costs 65? per gallon. It indicates that a savings for 14 mpg over 12 mpg is $116. One could ask the following ques tions: What is the equation of this surface =

f(x, y) where = savings in fuel costs for

y mpg over mpg? What are equations of curves of the surface in selected planes per pendicular to the xy plane? Have we stud ied the equations of some similar curves in the discussion above.?

Other Questions

A few additional practical questions of interest are the following: If a superior mo tor oil gives fifteen additional miles for

every twenty gallons of gasoline, how much

savings in dollars results per tankful, per year, and so on? Does the EPA guide in dicate that automatic transmissions in large cars provide the same level of fuel economy relative to manual transmissions as small cars? What are the comparative differences

in efficiency? How are insurance rates af fected by the differences in engine sizes?

By way of summary, a currently popular technique is to arrange instruction so that

mathematical concepts and algorithms arise out of real-life applications. The ex

amples in this article provide material for this type of instruction in the areas of lines,

hyperbolas, and rational and exponential functions. I hope you had some fun while

playing with these examples.

REFERENCES

Environmental Protection Agency (EPA). 1977 Gas

Mileage Guide. (Single copies can be obtained from Fuel Economy, Pueblo, Co. 81009. For bulk copies, write to Fuel Economy, Federal Energy Adminis

tration, DPM Room 6500, Washington, DC 20461.

Copies can also usually be obtained from new-car

dealerships.)

Skidell, Akiva. "The Harmonic Mean: A Nomograph, and Some Problems." Mathematics Teacher 70

(January 1977): 30-34.

Reader Reflections (coni, from p. 4) Of course, the use of Staib's proposed notation adds to its elegance. The proof of this theorem was accomplished inductively,

and was, to say the least, a laborious task.

John A. Price 8 Tray Drive Lititz, PA 17543

Editor's Note: John Staib responds: Your method of discov ery does show up in our article (see the italicized sentence on page 509 along with the figures and text immediately preced ing it). However, it shows up as an afterthought. The next time you prove your trinomial theorem in class, instead of using induction, try using a direct application of the binomial theo rem, as follows:

(x+y + zr = [x + (y + z)]m

- (

A Pascal source Perhaps some readers might be interested in the follow

ing reference, which is related to the article "The Pascal Pyramid" by John and Larry Staib (September 1978): Stephen Mueller, "Recursions Associated with Pascal's Pyramid," Pi Mu Epsilon Journal, Spring 1969, pp. 417-22. When Mueller wrote his article he was a student at what is now the University of Wisconsin?Oshkosh.

Robert W. Prielipp University of Wisconsin Oshkosh, WI 54901




Documents

SECONDARY SCHOOL MATHEMATICS FROM THE EPA GAS MILEAGE GUIDE