Stewart J. College Algebra: Concepts and Contexts

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204 CHAPTER 2

■

Linear Functions and Models

So the model we want is the linear equation .

(b) Since Mary’s total interest is $5025, we replace y by 5025 in the model

and solve the resulting one-variable linear equation.

Model

Replace y by 5025

Subtract 4500

Divide by 0.015 and switch sides

Calculator

So Mary invested $35,000 at 6% and the remaining $65,000 at .

■ NOW TRY EXERCISE 27 ■

Many real-world problems involve mixing different types of substances. For ex-

ample, construction workers may mix cement, gravel, and sand; fruit juice from con-

centrate may involve mixing different types of juices. Problems involving mixtures

and concentrations make use of the fact that if an amount x of a substance is dissolved

in a solution with volume , then the concentration C of the substance is given by

So if 10 grams of sugar are dissolved in 5 liters of water, then the sugar concentra-

tion is g/L.C = 10>5 = 2

C =

x = 35,000

x =

525

0.015

525 = 0.015x

5025 = 4500 + 0.015x

y = 4500 + 0.015x

= 4500 + 0.015x

= 0.06x + 4500 - 0.045x

y = 0.06x + 0.0451100,000 - x2

example

Constructing and Using a Model

(Mixtures and Concentration)

A manufacturer of soft drinks advertises its orange soda as “naturally flavored,” al-

though the soda contains only 5% orange juice. A new federal regulation stipulates

that to be called “natural,” a drink must contain at least 10% fruit juice. The manu-

facturer has a 900-gallon vat of soda and decides to add pure orange juice to the vat.

(a) Construct a model that gives the fraction of the mixture that is pure orange juice.

(b) How much pure orange juice must be added for the mixture to satisfy the

10% rule?

Solution

(a) Choose the variable. The variable in this problem is the amount of pure

orange juice added to the vat. So let

x = the amount 1in gallons2 of pure orange juice added

Distributive property

Simplify

Interest earned Interest earned

at 6% at %4

SECTION 2.6

■

Linear Equations: Getting Information from a Model 205

Translate words to algebra.

Let’s translate all the information given in the

problem into the language of algebra. First note that to begin with, 5% of the

900 gallons in the vat is orange juice, so the amount of orange juice in the vat

is gallons.10.05 2900 = 45

example

Constructing and Using a Model (Geometry)

Al paints with water colors on a sheet of paper that is 20 inches wide by 15 inches

high. He then places this sheet on a mat so that a uniformly wide strip of the mat

shows all around the picture.

(a) Construct a model that gives the perimeter of the mat.

(b) If the perimeter of the mat is 102 inches, how wide is the strip showing around

the picture?

Solution

(a) Choose the variable. The variable in this problem is the width of the strip.

So let

x = the width of the strip

In Words In Algebra

Amount of orange juice added x

Amount of the mixture

900 + x

Amount of orange juice in the mixture

45 + x

Set up the model. We are now ready to set up the model. To find the

fraction of the mixture that is orange juice, we divide the amount of orange

juice in the mixture by the amount of the mixture.

So the model we want is the equation .

(b) We want the mixture to have 10% orange juice. This means that we want the

fraction of the mixture that is orange juice to be 0.10. So we replace y by 0.10

in our model and solve the resulting one-variable linear equation for x.

Replace y by 0.10

Cross multiply

Distributive Property

Subtract 45, subtract 0.10x

Divide by 0.90 and switch sides

Calculator

So the manufacturer should add 50 gallons of pure orange juice to the soda.

■ NOW TRY EXERCISE 29 ■

x = 50

x =

0.90

45 = 0.90x

90 + 0.10x = 45 + x

0.101900 + x 2= 45 + x

0.10 =

45 + x

900 + x

y = 145 + x2>1900 + x2

y =

45 + x

900 + x

d Amount of orange

juice in the mixture

d Amount of the mixture

206 CHAPTER 2

■

Linear Functions and Models

Translate words to algebra. Figure 2 helps us to translate information

given in the problem into the language of algebra.

In Words In Algebra

Width of mat

20 + 2x

Length of mat

15 + 2x

20 in.

15 in.

figure 2

Set up the model. We are now ready to set up the model. To find the

perimeter of the mat, we add twice the width and twice the length.

Width Length

Distributive Property

Simplify

So the model we want is the equation .

(b) Since the perimeter is 102 inches, we replace y by 102 in our model and solve

the resulting one-variable equation for x.

Model

Replace y by 102

Subtract 70, switch sides

Divide by 8

Calculate

So if the perimeter is 102 inches, the width of the strip is 4 inches.

■ NOW TRY EXERCISE 31 ■

x = 4

x =

8 x = 32

102 = 70 + 8x

y = 70 + 8x

= 70 + 8x

= 40 + 4x + 30 + 4x

y = 2120 + 2x 2

+ 2115 + 2x 2

Check your knowledge of solving linear equations by doing the following prob-

lems. You can review these topics in

Algebra Toolkit C.1

on page T47.

1. Determine whether each given value is a solution of the equation.

(a) ; 2, 3 (b) ; 0, 6

2. Solve the given equation.

(a) (b) (c)

(d) (e)

(f)

3. Solve the given equation. Check that the solution satisfies the equation.

(a) (b) (c)

(d) (e)

2 +

x - 4

x + 1

x - 4

x + 1

x - 2

x + 3

- x - 2

2x - 2

x + 2

t + 6

t - 1

3x - 8

= 3

- 21x - 12= 512x + 32- 5x

3.2x + 1.4 = 10.9

+ 3 =

z + 7

31t - 82= 211 - 5t 2

1x - 82= 12x + 7 = 31

1, - 1

x + 6

x + 2

= 5

4x - 6

= 2x - 64x + 7 = 9x - 3

CONCEPTS

SECTION 2.6

■

Linear Equations: Getting Information from a Model 207

2.6 Exercises

Fundamentals

1. (a) The equation is a ______-variable equation.

(b) The equation is a

______-variable equation.

2. (a) To find the x-value when y is 6 in the equation , we solve the one-

variable equation

________.

(b) The solution of the one-variable equation in part (a) is

________.

3. Suppose the two-variable equation models the cost y of manufacturing

x flash drives. If the company spends $4000 on manufacturing flash drives, then to find

how many drives were manufactured, we solve the equation

________.

4. Alice invests $1000 in two certificates of deposit; the interest on the first certificate is

4%, and the interest on the second is 5%. Let’s denote the amount she invests in the first

certificate by x. Then the amount she invests in the second certificate is

__________.

The interest she receives on the first certificate is

__________, and the interest she

receives on the second certificate is

_________. So a function that models the total

interest she receives from both certificates is

_________.

Think About It

5–6 ■ True or false?

5. Bill is draining his aquarium. The equation models the volume y of water

remaining in the tank after x minutes. To find how long it takes for the tank to empty,

we replace y by 0 in the model and solve for x.

6. The equation models the profit y a street vendor makes from selling x

sandwiches. To find the number of sandwiches he must sell to make a profit of $45, we

replace x by 45 in the model and solve for y.

7–10

■ Find the value of x that satisfies the given equation when y is 4.

7. 8.

9. 10.

11–16 ■ The given equation models the relationship between the quantities x and y. Find

the value of x for the given value of y.

11. ;25 12. ; 100

13. ; 57.50 14. ; 1350

15. ; 0.5 16. ;8

17–20 ■ A quantity Q is related to a quantity R by the given equation. Find the value of Q

for the given value of R.

17. ;5 18. ;3

19. ; 20. ; - 5

R - 1

5 - 2Q

3Q - 4

- 2

Q - 5

2Q - 1

R - 2

= 412Q + 12

R + 1

= 21Q - 32

y =

5000 + 2x

250 + x

y =

10 + x

50 + x

y = 0.025x + 0.035150,000 - x2y = 0.05x + 0.0611000 - x2

y = 500 - 0.25xy = 5 + 2x

- 3y = 8 + 2x2x - 3y = 8

y = 6 - 4xy = 24 + 5x

y =-15 + 3x

y = 50 - 2x

y =

y = 500 + 10x

y - 2 = 3x + 1

3 = 8 + 5x

y = 8

+ 5x

SKILLS

208 CHAPTER 2

■

Linear Functions and Models

21. Weather Balloon A weather balloon is being filled. The linear equation

models the volume (in cubic feet) of hydrogen in the balloon at any time t (in

seconds). (See Exercise 55 in Section 2.2.) How many minutes will it take until the

balloon contains of hydrogen?

22. Filling a Pond A large koi pond is being filled. The linear equation

models the number y of gallons of water in the pool after x minutes. (See Exercise 56 in

Section 2.2.) If the pond has a capacity of 4000 gallons, how many minutes will it take

for the pond to be filled? Answer this question in two ways:

(a) Graphically (by graphing the equation and estimating the time from the graph)

(b) Algebraically (by solving an appropriate equation)

23. Landfill A county landfill has a maximum capacity of about 131,000,000 tons of

trash. The amount in the landfill on a given day since 1996 is modeled by the function

Here x is the number of days since January 1, 1996, and is measured in thousands

of tons. (See Exercise 53 in Section 2.2.) After how many days will the landfill reach

maximum capacity? Answer this question in two ways:

(a) Graphically (by graphing the equation and estimating the time from the graph)

(b) Algebraically (by solving an appropriate equation)

24. Air Traffic Controller An aircraft is approaching an international airport. Using radar,

an air traffic controller determines that the linear equation

models the distance (measured in miles) of the approaching aircraft from the radar

tower x minutes since the radar identified the aircraft. (See Exercise 57 in Section 2.3.)

How many minutes will it take for the aircraft to reach the radar tower?

25. Crickets and Temperature Biologists have observed that the chirping rate of a

certain species of cricket is modeled by the linear equation

where t is the temperature (in degrees Fahrenheit) and n is the number of chirps per

minute. (See Exercise 64 in Section 2.3.) If the temperature is , estimate the

cricket’s chirping rate (in chirps per minute).

26. Manufacturing Cost The manager of a furniture factory finds that the cost C (in

dollars) to manufacture x chairs is modeled by the linear equation

How many chairs are manufactured if the cost is $3500?

27. Investment Lili invests $12,000 in two one-year certificates of deposit. One

certificate pays 4%, and the other pays simple interest annually.

(a) Construct a model for the total interest Lili earns in one year on her investments.

(Let x represent the amount invested at 4%.)

(b) If Lili’s total interest is $526.00, how much money did she invest in each

certificate?

28. Investment Ronelio invests $20,000 in two one-year certificates of deposit. One

certificate pays 3%, and the other pays simple interest annually.3

C = 13x + 900

80°F

t =

n + 45

y =-4x + 45

T 1x 2

T 1x2= 32,400 + 4x

y = 300 + 10x

55 ft

V = 2 + 0.05t

CONTEXTS

SECTION 2.6

■

Linear Equations: Getting Information from a Model 209

(a) Construct a model for the total interest Ronelio earns in one year on his

investments. (Let x represent the amount invested at 3%.)

(b) If Ronelio’s total interest is $697.50, how much money did he invest in each

certificate?

29. Mixture A jeweler has five rings, each weighing 18 grams, made of an alloy of 10%

silver and 90% gold. He decides to melt down the rings and add enough silver to reduce

the gold content to 75%.

(a) Construct a model that gives the fraction of the new alloy that is pure gold. (Let x

represent the number of grams of silver added.)

(b) How much pure silver must be added for the mixture to have a gold content of 75%?

30. Mixture Monique has a pot that contains 6 liters of brine (salt water) at a

concentration of 120 g/L. She needs to boil some of the water off to increase the

concentration of salt.

(a) Construct a model that gives the concentration of the brine after boiling it. (Let x

represent the number of liters of water that is boiled off.)

(b) How much water must be boiled off to increase the concentration of the brine to

200 g/L?

31. Geometry A graphic artist needs to construct a design that uses a rectangle whose

length is 5 cm longer than its width x.

(a) Construct a model that gives the perimeter of the rectangle.

(b) If the perimeter of the rectangle is 26 cm, what are the dimensions of the rectangle?

32. Geometry An architect is designing a building whose footprint has the shape

shown below.

(a) Construct a model that gives the total area of the footprint of the building.

(b) Find x such that the area of the building is 144 square meters.

10 m

6 m

33. Law of the Lever The figure at left shows a lever system, similar to a seesaw that you

might find in a children’s playground. For the system to balance the product of the

weight and its distance from the fulcrum must be the same on each side; that is

This equation is called the Law of the Lever and was first discovered by Archimedes

(see page 568).

A mother and her son are playing on a seesaw. The boy is at one end, 8 ft from the

fulcrum. If the boy weights 100 lb and the mother weigh 125 lb, at what distance from

the fulcrum should the woman sit so that the seesaw is balanced?

34. Law of the Lever A 30 ft plank rests on top of a flat roofed building, with 5 ft of the

plank projecting over the edge, as shown in the figure. A 240 lb worker sits on one end

of the plank. What is the largest weight that can be hung on the projecting end of the

plank if it is to remain in balance? (Use the Law of the Lever as stated in Exercise 33.)

 w

„⁄

„¤

x⁄

x¤

5 ft

210 CHAPTER 2

■

Linear Functions and Models

2.7 Linear Equations: Where Lines Meet

■

Where Lines Meet

■

Modeling Supply and Demand

IN THIS SECTION… we learn how to find where the graphs of two linear functions

intersect. To find intersection points algebraically, we need to solve equations like the ones

we solved in the preceding section.

GET READY… by reviewing Section 1.7 on how to find graphically where the graphs of

two functions meet.

In many situations involving two linear functions we need to find the point where the

graphs of these functions intersect. In Section 1.7 we used the graphs of functions to

find their intersection points. In this section we will learn to use algebraic techniques

to find the intersection point of two linear functions.

■ Where Lines Meet

A well-known story from Aesop’s fables is about a race between a tortoise and a

hare. The hare is so confident of winning that he decides to take a long nap at the be-

ginning of the race. When he wakes up, he sees that the tortoise is very close to the

finish line; he realizes that he can’t catch up.

The hare’s confidence is justified; hares can sprint at speeds of up to 30 mi/h,

whereas tortoises can barely manage 0.20 mi/h. But let’s say that in this 2-mile race,

the tortoise tries really hard and keeps up a 0.8 mi/h pace; the complacent hare runs

at a leisurely 4 mi/h after a two and a quarter hour nap. Under these conditions, who

will win the race? Let’s see.

The tortoise starts the race at time zero, so the distance y he reaches in the race

course at time t is modeled by the equation or

Tortoise equation

Let’s find the linear equation that models the distance the hare reaches at time t. The

hare’s distance y is 0 when t is 2.25, so the point (2.25, 0) is on the graph of the de-

sired equation. Using the point-slope formula, we get , or

Hare equation

for . The graph in Figure 1 shows that the tortoise reaches the two-mile fin-

ish line before the hare does.

t Ú 2.25

y =-9 + 4t

y - 0 = 41t - 2.252

y = 0.8t

y = 0 + 0.8t

The hare is much faster than the

tortoise, but will he win the race?

Finish line

Hare overtakes

tortoise

Hare

Tortoise

figure 1 The hare and

tortoise race

SECTION 2.7

■

Linear Equations: Where Lines Meet 211

If the race is to continue beyond the finish line, when does the hare overtake the

tortoise? We can answer this question graphically (as in Section 1.7) or algebraically:

Graphically: From Figure 1 we estimate that the intersection point of the two

lines is approximately (2.8, 2.2). This point tells us that the hare overtakes the

tortoise about 2.8 hours into the race.

Algebraically: The hare and the tortoise have reached the same point in the

race course when the y-values (the distance traveled) in each equation are

equal. So we equate the y-values from each equation:

Solving for t, we get . So the rabbit overtakes the hare 2.8125 hours

into the race.

In general, the graphs of the functions f and intersect at those values of x for

which . In the special case in which the functions are linear, we have the

following.

Intersection Points of Linear Functions

f 1x 2= g1x2

t = 2.8125

0.8t =-9 + 4t

The accuracy of the graphical

method depends on how precisely

we can draw the graph. The

algebraic method always gives us

the exact answer.

To find where the graphs of the linear functions

intersect, we solve for x in the equation

+ m

x = b

+ m

y = b

+ m

xandy = b

+ m

example

Finding Where Two Lines Meet

Let and . Find the value of x where the graphs of f

and intersect, and find the point of intersection.

Solution 1 Graphical

We graph f and g in Figure 2. From the graph we see that the x-value of the intersec-

tion point is about x  5. Also from the graph we see that the y-value of the intersec-

tion point is more than 15, but it is difficult to see its exact value from the graph. So

from the graph we can estimate that the intersection point is (5, 15). (Note that this

answer is only an approximation and depends on how precisely we can draw the

graph.)

Solution 2 Algebraic

We solve the equation

Set functions equal

Add 8

Subtract 3x

Divide by 2

So when x  5. This means that the value of x where the graphs inter-

sect is 5.

f 1x 2= g 1x2

x = 5

2 x = 10

5 x = 10 + 3x

- 8 + 5x = 2 + 3x

g1x 2= 2 + 3xf 1x 2=-8 + 5x

figure 2 Graphs of f and g

212 CHAPTER 2

■

Linear Functions and Models

Let’s find the common value of f and when x  5:

So the point (5, 17) is on both graphs, thus the graphs intersect at (5, 17).

■ NOW TRY EXERCISE 5 ■

With rising fuel costs, there is an increasing interest in hybrid-electric vehicles,

which consume a lot less fuel than gas-powered vehicles do. Also, several manufac-

turers are now introducing fully electric cars, including some sporty models. The

Tesla fully electric sports car outperforms many gas-powered sports cars. The Tesla

can accelerate from 0 to 60 mi/h in 3.9 seconds. That matches the Lamborghini

Diablo’s acceleration. So now it is possible to have a fully electric car of your

dreams! Moreover, electric motors have few moving parts, so they require far less

maintenance (for example, the Tesla’s electric motor has only twelve moving parts).

g1x 2= 2 + 5132= 17

f 15 2=-8 + 5152=-8 + 25 = 17

IN CONTEXT ➤

Plug-in hybrid-electric car

Fully electric sports car

example

Cost Comparison of Gas-Powered and Hybrid-Electric Cars

Kevin needs to buy a new car. He compares the cost of owning two cars he likes:

Car A: An $18,000 gas-powered car that gets 20 mi/gal

Car B: A $25,000 hybrid-electric car that gets 48 mi/gal

For this comparison he assumes that the price of gas is $4.50 per gallon.

(a) Find a linear equation that models the cost of purchasing Car A and driving it

x miles.

(b) Find a linear equation that models the cost of purchasing Car B and driving it

x miles.

number of miles he needs to drive so that the cost of owning Car A is the same

as the cost of owning Car B.

Solution

(a) Since the gas-powered car gets 20 miles per gallon and the price of gas is

assumed to be $4.50, the cost of driving x miles is . Since4.501x>20 2= 0.225x

Jonathan Larsen/Shutterstock.com 2009

SECTION 2.7

■

Linear Equations: Where Lines Meet 213

example

Catching up with the Leader

Petra and Jordanna go on a bike ride from their home to the beach. Petra was slow

in getting ready to go, and she left the house 5 minutes later than Jordanna. On this

trip, Jordanna rides at an average speed of 5 m/s and Petra rides at an average speed

of 8 m/s. Let t  0 be the time Jordanna leaves home.

(a) Find a linear equation that models Jordanna’s distance y from home at time t.

(b) Find a linear equation that models Petra’s distance y from home at time t.

cycled when they meet?

Solution

(a) Since Jordanna cycles at an average speed of 5 m/s, the rate of change is

. For Jordanna, y is 0 when t is 0, so the initial value b is 0. So an

equation of the form that models the distance Jordanna has

traveled after t seconds is , or

Jordanna’s equation

(b) Petra’s average speed is 8 m/s, so the rate of change is . Petra leaves

later than Jordanna. So for Petra, y is 0

when t is 300. Thus the point (300, 0) is on the graph of the desired linear

equation. By the point-slope formula, the equation that models the distance

Petra has traveled after t seconds is , or

Petra’s equation

y-value in Petra’s equation. We set these two values equal to each other and

solve for t.

y = 8t - 2400

y - 0 = 81t - 3002

5 minutes = 5 * 60 = 300 seconds

m = 8

y = 5t

y = 5t + 0

y = mx + b

m = 5

the initial cost of this car is $18,000, the model for the cost of purchasing this

car and driving it x miles is

(b) Similarly, the model for the cost of purchasing the hybrid-electric car and

driving it x miles is

Set

Subtract 18,000

Subtract 0.09375x

Divide by 0.13125

So the hybrid-electric car needs to be driven about 53,333 miles before it

becomes cost-effective.

■ NOW TRY EXERCISE 27 ■

x L 53,333

0.13125x = 7,000

0.225x = 7,000 + 0.09375x

g1x 2= h 1x2 18,000 + 0.225x = 25,000 + 0.09375x

g1x 2= h 1x 2

h 1x2= 25,000 + 0.09375x

g1x 2= 18,000 + 0.225x