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3 Equations: Describing Relationships in Data

# 3 Equations: Describing Relationships in Data

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26

CHAPTER 1

Data, Functions, and Models

Dennis Sabo/Shutterstock.com 2009

For example, an ocean diver observes that the deeper she dives, the higher the

water pressure—she can feel the water pressing on her ears. How deep can she dive

before the pressure becomes dangerously high? To answer this question, she must be

able to predict what the pressure is at different depths without having to endanger

her life by diving to these depths. So she begins by diving to safe depths and measuring the water pressure. The data she obtains help her find a model (or equation)

that she can use to predict the pressure at depths to which she cannot possibly dive.

This situation is summarized as follows.

The data give the water pressure at different depths.

The model is an equation that represents the data.

Our goal is to use the model to predict the pressure at depths that are not in

the data.

The pressure-depth data and model are given below. Note that the single equation

P = 14.7 + 0.45d contains all the data and more. For instance, we can use this equation to predict the pressure at a depth of 200 ft, a value that is not available from the data.

Data

Depth

(ft)

Pressure

(lb/in2)

0

10

20

30

200

14.7

19.2

23.7

28.2

?

Model

Making a model

P = 14.7 + 0.45d

Using the model

In this section we learn how to make such models for data. The depth-pressure model

is obtained in Example 3.

2

■ Making a Linear Model from Data

A model is a mathematical representation (such as an equation) of a real-world situation. Modeling is the process of finding mathematical models. Once a model is

found, it can be used to answer questions about the thing being modeled.

Many real-world data start with an initial value for the output variable, and then

a fixed amount is added to the output variable for each unit increase in the input variable. For example, the production cost for manufacturing a certain number of cars consists of an initial fixed cost for setting up the equipment plus an additional unit cost for

manufacturing each car. In these cases we use a linear model to describe the data.

Linear Models

A linear model is an equation of the form y = A + Bx. In this model, A is the

initial value of y, that is, the value of y when x is zero, and B is the constant

amount by which y changes (increases or decreases) for each unit increase in x.

unit change in x

y =

r

Initial value

of y

r

Linear equations are studied in more

detail in Section 2.2.

A

+

Bx

SECTION 1.3

Equations: Describing Relationships in Data

27

In the next three examples we find some linear models from data.

e x a m p l e 1 Data, Equation, Graph

table 1

x

(chairs)

C

(dollars)

0

1

2

3

4

80

92

104

116

128

A furniture maker collects the data in Table 1, giving his cost C of producing x chairs.

(a) Find a linear model for the cost C of making x chairs.

(b) Draw a graph of the equation you found in part (a).

(c) What does the shape of the graph tell us about his cost of making chairs?

Solution

(a) The initial cost (the cost of producing zero chairs) is \$80. We can see from the

table that each chair produced costs an additional \$12. That is, the unit cost is

\$12. So an equation that models the relationship between C and x is

r

r

Initial cost

(or fixed cost) chair produced

C =      80      +       12x

✓ C H E C K To check that this equation correctly models the data, let’s try

some values for x. If four chairs are produced, then we can use the equation to

calculate the cost.

C = 80 + 12x

Model

C = 80 + 12142

Replace x by 4

= 128

Graphing equations is reviewed

in Algebra Toolkit D.2, page T71.

Calculate

This matches the cost given in the table for making four chairs. You can check

that the other values in the table also satisfy this equation.

(b) The ordered pairs in the table are solutions of the equation, so we plot them in

Figure 1(a). We can see that the points lie on a straight line, so we complete

the graph of the equation by drawing the line containing the plotted points as

in Figure 1(b).

C

150

140

130

120

110

100

90

80

0

C

150

140

130

120

110

100

90

80

1

2

3

4

5

(a) Graph from table

6 x

0

1

2

3

4

5

6 x

(b) Graph of equation

figure 1

(c) From the graph, it appears that cost increases steadily as the number of chairs

produced increases.

NOW TRY EXERCISES 7 AND 19

28

CHAPTER 1

Data, Functions, and Models

How can we tell whether a given set of data has a linear model? Let’s consider

data sets whose inputs are evenly spaced. For instance, the inputs 0, 1, 2, 3, . . . in

Example 1 are evenly spaced—successive inputs are one unit apart. For data with

evenly spaced inputs, there is a linear model for the data if the outputs increase (or

decrease) by a constant amount between successive inputs. We can test whether data

satisfy this condition by making a table of first differences. Each entry in the first

difference column of the table is the difference between an output and the immediately preceding output.

First Differences

For data with evenly spaced inputs:

The first differences are the differences in successive outputs.

If the first differences are constant, then there is a linear model for

the data.

The next example illustrates how we make and use a first difference table.

e x a m p l e 2 A Model for Temperature and Elevation

table 2

Elevation

(km)

Temperature

(°C)

0

1

2

3

4

5

20

10

0

- 10

- 20

- 30

10 - 20 = - 10

0 - 10 = - 10

- 10 - 0 = - 10

- 20 - 1- 102 = - 10

- 30 - 1- 202 = - 10

A mountain climber knows that the higher the elevation, the colder is the temperature. Table 2 gives data on the temperature at different elevations above ground level

on a certain day, gathered by using weather balloons.

(a) Show that a linear model is appropriate for these data.

(b) Find a linear model for the relationship between temperature and elevation.

(c) Draw a graph of the equation you found in part (b).

Solution

(a) We first observe that the inputs for these data are evenly spaced. To see

whether a linear model is appropriate, let’s make a table of first differences.

The entries in the first difference column are obtained by subtracting from

each output the preceding output.

Elevation

(km)

Temperature

(°C)

First

difference

0

1

2

3

4

5

20

10

0

- 10

- 20

- 30

- 10

- 10

- 10

- 10

- 10

We see that the first differences are constant (each is - 10), so there is a linear

model for these data.

(b) The linear model we seek is an equation of the form

T = A + Bh

where T represents temperature and h elevation.

SECTION 1.3

Equations: Describing Relationships in Data

29

When h is zero (ground level), the temperature is 20°C, so the initial value

A is 20. The first differences are the constant - 10, so the number B in the model

is - 10. We can now express the model as

r

Subtract 10 for each

kilometer of elevation

r

Temperature

at elevation 0

T =      20      -

10h

Notice how this equation fits the data. From the data we see that for each 1-km

increase in elevation, the temperature decreases by 10°C. So at an elevation of

h km we must subtract 10h degrees from the ground temperature.

✓ C H E C K To check that this equation correctly models the data, we try

some values for h from the table. For example, if the elevation is 5 km, we replace h by 5 in the model:

T

20

10

0

_10

1

2

3

4

5

6 h

_20

_30

_40

figure 2

Graph of T = 20 - 10h

T = 20 - 10h

Model

T = 20 - 10152

Replace h by 5

T = - 30

Calculate

This matches the temperature given in the table for an elevation of 5 km. You

can check that the other values in the table also satisfy this equation.

(c) We plot the points in the table and then complete the graph by drawing the

line that contains the plotted points. (See Figure 2.)

NOW TRY EXERCISES 11 AND 21

e x a m p l e 3 A Model for Depth-Pressure Data

table 3

Depth

(ft)

Pressure

(lb/in2)

0

10

20

30

40

50

14.7

19.2

23.7

28.2

32.7

37.2

A scuba diver obtains the depth-pressure data shown in Table 3.

(a) Show that a linear model is appropriate for these data.

(b) Find a linear model that describes the relationship between depth and pressure.

Solution

(a) The inputs for these data are evenly spaced. We make a table of first differences.

Depth

(ft)

Pressure

(lb/in2)

First

differences

0

10

20

30

40

50

14.7

19.2

23.7

28.2

32.7

37.2

4.5

4.5

4.5

4.5

4.5

We see that the first differences are constant (each is 4.5), so there is a linear

model for these data.

(b) The linear model we seek is an equation of the form

P = A + Bd

where P represents pressure and d represents depth.

30

CHAPTER 1

Data, Functions, and Models

When the depth d is 0, the pressure is 14.7 lb/in2, so the initial value A is

14.7. From the first difference column in the table we see that pressure increases

by 4.5 lb/in2 for each 10-ft increase in depth. So for each 1-ft increase in depth

the pressure increases by

4.5

= 0.45 lb/in 2

10

Pressure at

depth 0

each foot of depth

r

r

So the number B in the model is 0.45. We can now express the model as

P = 14.7      +        0.45d

If d is 0, then P = 14.7 + 0.4510 2 = 14.7.

If d is 10, then P = 14.7 + 0.45110 2 = 19.2.

If d is 30, then P = 14.7 + 0.45130 2 = 28.2.

✓ C H E C K Let’s check whether this model fits the data. For instance, when

d is 20, we get P = 14.7 + 0.451202 = 23.7, which agrees with the table. In the

margin we check the model against other entries in the table.

2

■ Getting Information from a Linear Model

IN CONTEXT ➤

NOW TRY EXERCISE 23

The point of making a model is to use it to predict conditions that are not directly observed in our data.

In the next example we use the depth-pressure model of Example 3 to find the

pressure at depths to which no human can dive unaided. This illustrates the power of

the modeling process: It allows us to explore properties of the real world that are beyond our physical experience. In the first-ever attempt to explore the ocean depths,

Otis Barton and William Beebe built a steel sphere (see the photo) with a diameter

of 4 ft 9 in., which they called the bathysphere (bathys is the Greek word for deep).

They needed to build their craft to be strong enough to withstand the crushing water

pressure at the great depths to which they planned to descend. They used the depthpressure model to estimate the pressure at those depths and then built the bathysphere accordingly. On August 15, 1934, they successfully descended to a depth of

3028 ft below the surface of the Atlantic. From the bathysphere’s portholes they observed exciting new marine species that had never before been seen by humans.

e x a m p l e 4 Using the Depth-Pressure Model

The bathysphere described above is lowered to the bottom of a deep ocean trench.

Use the depth-pressure model P = 14.7 + 0.45d to predict the pressure at a depth

of 3000 ft.

Solution

Since the depth is 3000 ft, we replace d by 3000 in the model and solve for P:

P = 14.7 + 0.45d

Model

P = 14.7 + 0.45130002

Replace d by 3000

P = 1364.7

Calculate

2

So the pressure is 1364.7 lb/in .

NOW TRY EXERCISE 25

SECTION 1.3

Equations: Describing Relationships in Data

31

Test your skill in graphing equations in two variables. You can review this topic

in Algebra Toolkit D.2 on page T71.

1. An equation is given. Determine whether the given point (x, y) is a solution

of the equation.

(a) y = 5x - 3; (3, 12)

(c) y = 25 - 3x 3; 1- 2, 12

(b) 5x - 2y = 4; (2, 3)

2. An equation is given. Determine whether the given point (x, y) is on the

graph of the equation.

(a) y = 3x - 7; 12, - 1 2

(c) y = 2x 2 + 1; 1- 1, 32

(b) 3y - x = - 6; (2, 0)

3. An equation is given. Complete the table and graph the equation.

(a) y = 2 + x

x

0

y

2

(b) y = 1 - 3x

1

2

3

4

5

(c) y = - 7 + 4x

x

0

y

-7

1

x

-2

y

7

-1

0

1

2

3

(d) y = - 3 - 2x

2

3

4

5

x

-3

y

3

-2

-1

0

1

2

4. Find the x- and y-intercepts of the given equation.

(a) y = x - 2

(b) 2y = 3x + 2

(c) 2x - 6y = 12

(d) 4x - 5y = 8

5. Graph the given equation, and find the x- and y-intercepts.

(a) y = 2x

(b) y = 6 + 2x

(c) y = - 6 + 3x

(d) 3y = 6 - 2x

1.3 Exercises

CONCEPTS

Fundamentals

1. (a) What is a model? Give some examples of models you use every day.

(b) If you work for \$15 an hour, describe the relation between your pay and the number

of hours you work, using (i) a table, (ii) a graph, (iii) an equation.

x

y

First difference

0

45

1

39

2

33

3

27

4

21

5

15

2. For data with evenly spaced inputs, if the first differences are _______, then a linear

model is appropriate for the data. In the data shown in the margin, x represents the input

and y represents the output. What are the first differences? Is a linear model appropriate?

3. The equation L = 4S is a linear model for the total number of legs L that S sheep have.

Using the model, we find that 12 sheep have L = 4 *

legs.

=

ٗ ٗ

4. Suppose digital cable service costs \$49 a month with an initial installation fee of \$110.

We make a linear model for the total cost C of digital cable service for x months by

writing the equation ______________.

5. What is the purpose of making a model? Support your answer by examples.

6. Explain how data, equations, and graphs work together to describe a real-world situation.

Give an example of a real-world situation that can be described in these three ways.

32

CHAPTER 1

SKILLS

Data, Functions, and Models

7–10 ■

(a)

(b)

(c)

7.

A set of data is given.

Find a linear model for the data.

Use the model to complete the table.

Draw a graph of the model.

x

y

0

8.

9.

10.

a

b

55

0

- 10

1

52

1

-4

86

2

49

2

2

74

3

46

3

8

u

v

A

B

5

0

110

0

1

12

1

98

2

19

2

3

26

3

4

4

4

4

5

5

5

5

6

6

6

6

11–14 ■ A set of data is given.

(a) Find the first differences.

(b) Is a linear model appropriate? If so, find a linear model for the data.

(c) If there is a linear model, use it to complete the table.

11.

First

difference

12.

x

y

205

0

60

1

218

1

54

2

231

2

48

3

244

3

42

x

y

0

4

4

5

5

6

6

13.

First

difference

14.

x

y

23

0

17

1

19

1

38

2

16

2

59

3

11

3

80

x

y

0

4

4

5

5

6

6

First

difference

First

difference

SECTION 1.3

15–18

15.

Equations: Describing Relationships in Data

Find a linear model for the data graphed in the scatter plot.

y

30

16.

20

10

0

17.

1

Tony Ayling

CONTEXTS

Male and female angler fish

2

18.

0.1

0.2

0.3

0.4 x

y

100

80

60

40

20

0

4 x

3

y

1000

800

600

400

200

0

33

1

2

3

4 x

y

2.50

2.00

1.50

1.00

0.50

0

5 10 15 20 25 30 35 40 x

19. Truck Rental A home improvement store provides short-term truck rentals for their

customers to take large items home. The store charges a base rate of \$19 plus a time

charge for every half hour that the truck is used. The table gives rental rate data for

different rental periods.

(a) Find a linear model for the relation between the rental cost and rental period (in hours).

(b) Draw a graph of the equation you found.

(c) Use the model to predict the rental cost for 5 hours.

Period (h)

Rental cost (\$)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

19.00

24.00

29.00

34.00

39.00

44.00

49.00

20. Aquatic Life in the Midnight Zone In Example 6 we used the model

P = 14.7 + 0.45d to find the pressure (lb/in2) at various ocean depths. The deepest

ocean trenches plunge to an astounding 7 miles below sea level, far too deep for

sunlight to penetrate. Yet our planet is so teeming with life that even at these depths

there are living creatures. Anglerfish can live at depths up to 11,000 ft and are

characterized by luminescent appendages, which they use to lure their prey. The

scientific name of the bizarre-looking anglerfish shown here, linophryne arborifera,

means, roughly, “toad that fishes with a tree-like net.”

(a) Use the model to predict the pressure at the 11,000-ft depth where anglerfish can live.

(b) One atmosphere (atm) is defined as a pressure of 14.7 lb/in2, which is the normal

air pressure we experience at sea level. Convert your answer in part (a) to

atmospheres. How many times greater is the pressure under which anglerfish live

than the pressure under which we live?

21. Boiling Point Most high-altitude hikers know that cooking takes longer at higher

elevations. This is because the atmospheric pressure decreases as the elevation increases,

causing water to boil at a lower temperature, and food cooks more slowly at that lower

34

CHAPTER 1

Data, Functions, and Models

Peter Zaharov/Shutterstock.com 2009

temperature. The table below gives data for the boiling point of water at different elevations.

(a) Use first differences to show that a linear model is appropriate for the data.

(b) Find a linear model for the relation between boiling point and elevation.

(c) Use the model to predict the boiling point of water at the peak of Mount

Kilimanjaro, 19,340 ft above sea level.

Mount Kilimanjaro

Elevation

(؋ 1000 ft)

Boiling point

(°F)

First

difference

0

212.0

1

210.2

2

208.4

3

206.6

4

204.8

5

203.0

22. Temperatures on Mount Kilimanjaro Mount Kilimanjaro is the highest mountain

in Africa. Its snow-covered peak rises 4800 m above the surrounding plain. It is located

in northern Tanzania near the equator, and conditions on the mountain vary from

equatorial, to tropical, to arctic, because of the steadily decreasing temperature as the

altitude increases. The table below gives data for the temperature on a typical day on

Kilimanjaro at various elevations above the base of the mountain.

(a) Use first differences to show that a linear model is appropriate for the data.

(b) Find a linear model for the relation between the temperature on Kilimanjaro and

the elevation above the base of the mountain.

(c) Use the model to predict what the temperature will be on a typical day at the peak

of Kilimanjaro.

Elevation

above base (m)

Miles

driven

Pounds of

chocolate used

0

20

40

60

80

100

0

17

34

51

68

85

Temperature

(°C)

First

difference

0

30

400

28

800

26

1200

24

1600

22

2000

20

23. Chocolate-Powered Car Two British entrepreneurs, Andy Pag and John Grimshaw,

drove 4500 miles from England to Timbuktu, Mali, in a truck powered by chocolate. They

used an ethanol that is made from old, unusable chocolate, and it took about 17 pounds of

chocolate to make 1 gallon of ethanol. The table in the margin gives data for the

relationship between the amount of chocolate used and the number of miles driven.

(a) Use first differences to show that a linear model is appropriate for the data.

(b) Find a linear model for the relation between the amount of chocolate used and the

number of miles driven.

(c) Use the model found in part (b) to predict how many pounds of chocolate it took to

drive from England to Timbuktu.

SECTION 1.4

Profit (\$)

0

10

20

30

40

50

- 80.00

- 40.00

0.00

40.00

80.00

120.00

Salary (ϫ \$1000)

Number

Books (thousands)

2 3

Year

4

5 x

y

40

30

20

10

0

2

1

1

Functions: Describing Change

35

24. Profit An Internet company sells cell phone accessories. The table in the margin gives

the profit they make on selling battery chargers. (Note that negative numbers in the

table represent a loss.) Because of storage costs, the company needs to sell at least 20

chargers before they begin to make a profit.

(a) Find a linear model for the relation between profit and the number of battery

chargers sold.

(b) Draw a graph of the equation you found.

(c) Use the model to predict the profit from selling 150 battery chargers.

y

200

190

180

170

160

150

0

2 3 4 5 6 x

Years since 2001

25. Salary A woman is hired as CEO of a small company and is offered a salary of

\$150,000 for the first year. In addition, she is promised regular salary increases. The

graph in the margin shows her potential salary (in thousands of dollars) for the first few

years that she works for the company.

(a) Find a linear model for the relation between her salary and the number of years she

works for the company.

(b) Use the model to predict what her salary will be after she has worked 10 years for

the company.

26. Library Book Collection A city library remodeled and expanded in the year 2001

and increased its maximum capacity to about 100,000 books. In 2001 the library held

about 20,000 books, and each subsequent year the library adds a fixed number of books

to its collection. The graph in the margin plots the number of books (in thousands) the

library held each year from 2001 to 2007.

(a) Find a linear model for the relation between the number of books in the library and

the number of years since 2001.

(b) Use the model to predict the number of books in the library after 25 years (in 2026).

1.4 Functions: Describing Change

Definition of Function

Which Two-Variable Data Represent Functions?

Which Equations Represent Functions?

Which Graphs Represent Functions?

Four Ways to Represent a Function

IN THIS SECTION… we begin our study of functions. There are four basic ways to

represent functions: words, data, equations, and graphs. The concept of function is a

versatile tool for modeling the real world. In succeeding chapters we study more properties

of functions; each new property provides a new modeling tool.

GET READY… by reviewing how to solve equations in Algebra Toolkit C.1. Test your

skill by doing the Algebra Checkpoint at the end of this section.

In Sections 1.1–1.3 we saw how analyzing two-variable data can reveal relationships

between the variables. Such relationships can be seen from the data themselves, from

a graph, or from an equation. But this is not the whole story; in many real-world situations we are interested in how a change in one variable results in a change in the

other variable. We’ll study these types of relations using the concept of function.

Equipped with this concept, we will be able make a great leap in our understanding

of our ever-changing world.

36

2

CHAPTER 1

Data, Functions, and Models

■ Definition of Function

We use the term function to describe the dependence of one changing quantity on another. For example, we say that the height of a child is a function of the child’s age,

the weather is a function of the date, the cost of mailing a package is a function of its

weight, and so on. Situations like these, which involve change, have the property that

each input (or each value for the first variable) results in exactly one output (exactly

one value for the other variable). For instance, mailing a package does not result in

two different costs. This leads us to the following definition of function.

Definition of Function

A function is a relation in which each input gives exactly one output.

We can easily tell whether a relation is a function by making a diagram, as we did in

Section 1.2. The diagram in Figure 1(a) represents a function because for each input

there is exactly one output. But the relation described by the diagram in Figure 1(b)

is not a function—the input 2 corresponds to two different outputs, 20 and 30.

10

20

30

40

1

2

3

4

Inputs

Outputs

10

20

30

40

50

1

2

3

4

Inputs

(a) A function

Outputs

(b) Not a function

f i g u r e 1 When is a relation a function?

e x a m p l e 1 Which Relations Are Functions?

A relation is given by a table. The input is in the first column, and the output in the

second column. Is the relation a function?

(a) Table 1 gives the number of women in the U.S. Senate between 1997 and 2007.

(b) Table 2 gives the ages of women in a certain neighborhood and the number of

children each woman has.

table 1

table 2

Women in U.S. Senate

Number of children

Year

Number

Age

Number

1997

1999

2001

2003

2005

2007

9

9

14

14

14

16

31

32

24

35

31

22

3

0

1

1

2

0

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