section 1.3 models and applications

116 Chapter 1 Equations and Inequalities

125. The equations and are equivalent.

126. The equations and are equivalent.

127. If and are any real numbers, then always hasone number in its solution set.

128. If represents a number, write an English sentence aboutthe number that results in an inconsistent equation.

129. Find such that has a solution set givenby

130. Find such that has a solution set given by �.4x - b

x - 5= 3b

5-66.

7x + 4b

+ 13 = xb

x

ax + b = 0ba

3y - 7 = 53y - 1 = 11

x = 4x

x - 4=

4x - 4

Preview ExercisesExercises 131–133 will help you prepare for the material coveredin the next section.

131. Jane’s salary exceeds Jim’s by $150 per week. If representsJim’s weekly salary, write an algebraic expression thatmodels Jane’s weekly salary.

132. A long-distance telephone plan has a monthly fee of $20with a charge of $0.05 per minute for all long-distance calls.Write an algebraic expression that models the plan’smonthly cost for minutes of long-distance calls.

133. If the width of a rectangle is represented by and thelength is represented by write a simplifiedalgebraic expression that models the rectangle’s perimeter.

x + 200,x

x

x

1.3 Models and Applications

In this section, you’ll see examples and exercises focused on how much moneyAmericans earn. These situations illustrate a step-by-step strategy for solving

problems. As you become familiar with this strategy, you will learn to solve a widevariety of problems.

Problem Solving with Linear EquationsWe have seen that a model is a mathematical representation of a real-worldsituation. In this section, we will be solving problems that are presented inEnglish. This means that we must obtain models by translating from the ordinarylanguage of English into the language of algebraic equations. To translate,however, we must understand the English prose and be familiar with the forms ofalgebraic language. On the next page are some general steps we will follow insolving word problems:

Objectives

� Use linear equations to solveproblems.

� Solve a formula for a variable.

S e c t i o n

How Long It Takes to Earn $1000

Howard SternRadio host

24 sec.

Dr. Phil McGrawTelevision host2 min. 24 sec.

Brad PittActor

4 min. 48 sec.

Kobe BryantBasketball player

5 min. 30 sec.

Chief executiveU.S. average2 hr. 55 min.

Doctor, G.P.U.S. average13 hr. 5 min.

High school teacherU.S. average

43 hours

JanitorU.S. average

103 hours

Source: Time

� Use linear equations to solveproblems.

C-BLTZMC01_089-196-hr2 12-09-2008 10:39 16

Section 1.3 Models and Applications 117

Strategy for Solving Word Problems

Step 1 Read the problem carefully. Attempt to state the problem in your ownwords and state what the problem is looking for. Let (or any variable)represent one of the unknown quantities in the problem.

Step 2 If necessary, write expressions for any other unknown quantities in theproblem in terms of

Step 3 Write an equation in that models the verbal conditions of the problem.

Step 4 Solve the equation and answer the problem’s question.

Step 5 Check the solution in the original wording of the problem, not in theequation obtained from the words.

x

x.

x

Study TipWhen solving word problems,particularly problems involvinggeometric figures, drawing a pictureof the situation is often helpful.Label on your drawing and, whereappropriate, label other parts of thedrawing in terms of x.

x

Celebrity Earnings

Forbes magazine published a list of the highest paid TV celebrities between June2006 and June 2007. The results are shown in Figure 1.15.

EXAMPLE 1

Highest Paid TV Celebrities between June 2006 and June 2007

OprahWinfrey

JerrySeinfeld

SimonCowell

DavidLetterman

DonaldTrump

0

40

80

120

160

200

240

280

320

Celebrity

360

400

440

Ear

ning

s (m

illio

ns o

f dol

lars

)

$40 million $32 million

Figure 1.15Source: Forbes

The bar heights indicate that nobody came close to Oprah, who earned over fourtimes more than any of the other TV stars. Although Seinfeld earned $15 millionmore than Cowell, Oprah’s earnings exceeded Cowell’s by $215 million. Combined,these three celebrities earned $365 million. How much did each of them earn?

SolutionStep 1 Let represent one of the unknown quantities. We know something aboutSeinfeld’s earnings and Oprah’s earnings: Seinfeld earned $15 million more thanCowell, and Oprah’s earnings exceeded Cowell’s by $215 million. We will let

Step 2 Represent other unknown quantities in terms of Because Seinfeldearned $15 million more than Cowell, let

Because Oprah’s earnings exceeded Cowell’s by $215 million, let

x + 215 = Oprah’s earnings.

x + 15 = Seinfeld’s earnings.

x.

x = Cowell’s earnings 1in millions of dollars2.

x

C-BLTZMC01_089-196-hr2 12-09-2008 10:39 Page 117

Study TipModeling with the word exceeds can be a bit tricky. It’s helpful to identify the smallerquantity.Then add to this quantity to represent the larger quantity. For example, suppose thatTim’s height exceeds Tom’s height by inches. Tom is the shorter person. If Tom’s height isrepresented by then Tim’s height is represented by x + a.x,

a


Step 3 Write an equation in that models the conditions. Combined, the threecelebrities earned $365 million.

Step 4 Solve the equation and answer the question.This is the equation that modelsthe problem’s conditions.Remove parentheses, regroup,and combine like terms.Subtract 230 from both sides.Divide both sides by 3.

Thus,

Between June 2006 and June 2007, Oprah earned $260 million, Seinfeld earned$60 million, and Cowell earned $45 million.

Step 5 Check the proposed solution in the original wording of the problem. Theproblem states that combined, the three celebrities earned $365 million. Using theearnings we determined in step 4, the sum is

or $365 million, which verifies the problem’s conditions.

$45 million + $60 million + $260 million,

Oprah’s earnings = x + 215 = 45 + 215 = 260. Seinfeld’s earnings = x + 15 = 45 + 15 = 60

Cowell’s earnings = x = 45

x = 45 3x = 135

3x + 230 = 365

1x + 2152 + 1x + 152 + x = 365

(x+215) + (x+15) + x = 365

Oprah’s earnings Seinfeld’s earningsCowell’searningsplus plus equal $365 million.

x

Women

Men

Figure 1.16Source: John Macionis, Sociology, Twelfth Edition, Prentice Hall, 2008

Check Point 1 According to the U.S. Department of Education (2007 data),there is a gap between teaching salaries for men and women at private collegesand universities. The average salary for men exceeds the average salary forwomen by $14,037. Combined, their average salaries are $130,015. Determine theaverage teaching salaries at private colleges for women and for men.

Modeling Attitudes of College Freshmen

Researchers have surveyed college freshmenevery year since 1969. Figure 1.16 shows thatattitudes about some life goals have changeddramatically. Figure 1.16 shows that thefreshmen class of 2006 was more interested inmaking money than the freshmen of 1969 hadbeen. In 1969, 52% of first-year college menconsidered “being very well off financially”essential or very important. For the period from1969 through 2006, this percentage increased byapproximately 0.6 each year. If this trend

EXAMPLE 2

Life Objectives of College Freshmen, 1969–2006

“Being very welloff financially”

Life Objective

“Developing a meaningfulphilosophy of life”

60%

20%

40%

Women Men

80%

100%

50%

10%

30%

70%

90%

Per

cent

age

Cal

ling

Obj

ecti

ve“E

ssen

tial

” or

“V

ery

Impo

rtan

t”

1969

52%

30%

2006

75%72%

1969

82%88%

2006

47%46%

C-BLTZMC01_089-196-hr2 12-09-2008 10:39 Page 118


continues, by which year will all male freshmen consider “being very well offfinancially” essential or very important?

SolutionStep 1 Let represent one of the unknown quantities. We are interested in theyear when all male freshmen, or 100% of the men, will consider this life objectiveessential or very important. Let

Step 2 Represent other unknown quantities in terms of There are no otherunknown quantities to find, so we can skip this step.

Step 3 Write an equation in that models the conditions.

Step 4 Solve the equation and answer the question.

This is the equation that modelsthe problem’s conditions.

Subtract 52 from both sides.

Simplify.

Divide both sides by 0.6.

Simplify.

Using current trends, by 80 years after 1969, or in 2049, all male freshmen willconsider “being very well off financially” essential or very important. (Do you agreewith this projection that extends so far into the future? Are there unexpected eventsthat might cause model breakdown to occur?)

Step 5 Check the proposed solution in the original wording of the problem. Theproblem states that all men (100%, represented by 100 using the model) willconsider the objective essential or very important. Does this occur if we increase the1969 percentage, 52%, by 0.6 each year for 80 years, our proposed solution?

This verifies that using trends shown in Figure 1.16, all first-year college men willconsider the objective essential or very important 80 years after 1969.

Check Point 2 Figure 1.16 shows that the freshmen class of 2006 was lessinterested in developing a philosophy of life than the freshmen of 1969 hadbeen. In 1969, 88% of the women considered this objective essential or veryimportant. Since then, this percentage has decreased by approximately 1.1each year. If this trend continues, by which year will only 33% of femalefreshmen consider “developing a meaningful philosophy of life” essential orvery important?

52 + 0.61802 = 52 + 48 = 100

x = 80

0.6x

0.6=

480.6

0.6x = 48

52 - 52 + 0.6x = 100 - 52

52 + 0.6x = 100

52 + 0.6x = 100

The 1969percentage

0.6 each yearfor x years

100% of themale freshmen.

increasedby equals

x

x.

x = the number of years after 1969 when allmale freshmen will consider “being verywell off financially” essential or veryimportant.

x

C-BLTZMC01_089-196-hr2 12-09-2008 10:39 Page 119


Selecting a Long-Distance Carrier

You are choosing between two long-distance telephone plans. Plan A has a monthlyfee of $20 with a charge of $0.05 per minute for all long-distance calls. Plan B has amonthly fee of $5 with a charge of $0.10 per minute for all long-distance calls. Forhow many minutes of long-distance calls will the costs for the two plans be thesame?

SolutionStep 1 Let represent one of the unknown quantities. Let

Step 2 Represent other unknown quantities in terms of There are no otherunknown quantities, so we can skip this step.

Step 3 Write an equation in that models the conditions. The monthly cost forplan A is the monthly fee, $20, plus the per minute charge, $0.05, times the numberof minutes of long-distance calls, The monthly cost for plan B is the monthly fee,$5, plus the per-minute charge, $0.10, times the number of minutes of long-distancecalls,



Subtract from both sides.



Simplify.

Because represents the number of minutes of long-distance calls for whichthe two plans cost the same, the costs will be the same for 300 minutes of long-distance calls.

Step 5 Check the proposed solution in the original wording of the problem. Theproblem states that the costs for the two plans should be the same. Let’s see if theyare the same with 300 minutes of long-distance calls:

With 300 minutes, or 5 hours, of long-distance chatting, both plans cost $35 for themonth. Thus, the proposed solution, 300 minutes, satisfies the problem’sconditions.

Per-minute chargeMonthly fee

Cost for plan A=$20+$0.05(300)=$20+$15=$35

Cost for plan B=$5+$0.10(300)=$5+$30=$35.

x

300 = x

15

0.05=

0.05x

0.05

15 = 0.05x

0.05x 20 = 5 + 0.05x

20 + 0.05x = 5 + 0.10x

The monthlycost for plan A

the monthlycost for plan B.

mustequal

20+0.05x = 5+0.10x

x.

x.

x

x.

x = the number of minutes of long-distance callsfor which the two plans cost the same.

x

EXAMPLE 3

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Check Point 3 You are choosing between two long-distance telephone plans.Plan A has a monthly fee of $15 with a charge of $0.08 per minute for all long-distance calls. Plan B has a monthly fee of $3 with a charge of $0.12 per minute forall long-distance calls. For how many minutes of long-distance calls will the costsfor the two plans be the same?

A Price Reduction on a Digital Camera

Your local computer store is having a terrific sale on digital cameras. After a 40%price reduction, you purchase a digital camera for $276.What was the camera’s pricebefore the reduction?

SolutionStep 1 Let represent one of the unknown quantities. We will let

Step 2 Represent other unknown quantities in terms of There are no otherunknown quantities to find, so we can skip this step.

Step 3 Write an equation in that models the conditions. The camera’s originalprice minus the 40% reduction is the reduced price, $276.

Step 4 Solve the equation and answer the question.This is the equation that models the problem’sconditions.Combine like terms:


The digital camera’s price before the reduction was $460.

Simplify: 0.6 276.0460.

� x = 460

0.6x

0.6=

2760.6

x - 0.4x = 1x - 0.4x = 0.6x. 0.6x = 276

x - 0.4x = 276

minus isOriginalprice

the reduction(40% of the

original price)the reducedprice, $276.

x - 0.4x = 276

x

x.

x = the original price of the digital camera prior to the reduction.x

EXAMPLE 4

TechnologyNumeric and Graphic Connections

We can use a graphing utility to numerically or graphically verify our work in Example 3.

The monthlycost for plan A

the monthlycost for plan B.

mustequal

20+0.05x = 5+0.10x

Enter y1 = 20 + .05x. Enter y2 = 5 + .10x.

Numeric Check Graphic Check

Display a table for and

When x = 300, y1 and y2 havethe same value, 35. With 300minutes of calls, costs are the

same, $35, for both plans.

y2 .y1 Display graphs for and Use the intersection feature.

Graphs intersect at (300, 35).With 300 minutes of calls,costs are the same, $35, for

both plans.

y1 = 20 + 0.05x

y2 = 5 + 0.10x

[0, 500, 100] by [0, 50, 5]

y2 .y1

Study TipObserve that the original price,reduced by 40% is and notx - 0.4.

x - 0.4xx,

C-BLTZMC01_089-196-hr2 12-09-2008 10:39 Page 121


Step 5 Check the proposed solution in the original wording of the problem. Theprice before the reduction, $460, minus the 40% reduction should equal the reducedprice given in the original wording, $276:

This verifies that the digital camera’s price before the reduction was $460.

Check Point 4 After a 30% price reduction, you purchase a new computer for$840. What was the computer’s price before the reduction?

Our next example is about simple interest. Simple interest involves interestcalculated only on the amount of money that we invest, called the principal. Theformula is used to find the simple interest, earned for one year when theprincipal, is invested at an annual interest rate, Dual investment problemsinvolve different amounts of money in two or more investments, each paying adifferent rate.

Solving a Dual Investment Problem

Your grandmother needs your help. She has $50,000 to invest. Part of this money isto be invested in noninsured bonds paying 15% annual interest. The rest of thismoney is to be invested in a government-insured certificate of deposit paying7% annual interest. She told you that she requires $6000 per year in extra incomefrom the combination of these investments. How much money should be placed ineach investment?

SolutionStep 1 Let represent one of the unknown quantities. We will let

Step 2 Represent other unknown quantities in terms of The other quantity thatwe seek is the amount invested at 7% in the certificate of deposit. Because the totalamount Grandma has to invest is $50,000 and we already used up

Step 3 Write an equation in that models the conditions. Because Grandmarequires $6000 in total interest, the interest for the two investments combined mustbe $6000. Interest is or for each investment.



Use the distributive property.

Combine like terms.



Simplify. x = 31,250

0.08x

0.08=

25000.08

0.08x = 2500

0.08x + 3500 = 6000

0.15x + 3500 - 0.07x = 6000

0.15x+0.07(50,000-x)=6000

0.15x + 0.07(50,000-x) = 6000

rate times principal rate times principal

Interest from the15% investment

interest from the7% investmentplus is $6000.

rPPr

x

50,000 - x = the amount invested in the certificate of deposit at 7%.

x,

x.

x = the amount invested in the noninsured bonds at 15%.

x

EXAMPLE 5

r.P,I,I = Pr

460 - 40% of 460 = 460 - 0.414602 = 460 - 184 = 276.

C-BLTZMC01_089-196-hr2 12-09-2008 10:39 Page 122


Thus,

Grandma should invest $31,250 at 15% and $18,750 at 7%.

Step 5 Check the proposed solution in the original wording of the problem. Theproblem states that the total interest from the dual investments should be $6000.Can Grandma count on $6000 interest? The interest earned on $31,250 at 15% is($31,250) (0.15), or $4687.50. The interest earned on $18,750 at 7% is($18,750) (0.07), or $1312.50. The total interest is or $6000,exactly as it should be. You’ve made your grandmother happy. (Now if you wouldjust visit her more often )

Check Point 5 You inherited $5000 with the stipulation that for the first year themoney had to be invested in two funds paying 9% and 11% annual interest. Howmuch did you invest at each rate if the total interest earned for the year was $487?

Solving geometry problems usually requires a knowledge of basic geometricideas and formulas. Formulas for area, perimeter, and volume are given in Table 1.1.

Á

$4687.50 + $1312.50,

the amount invested at 7% = 50,000 - 31,250 = 18,750.

the amount invested at 15% = x = 31,250.

Table 1.1 Common Formulas for Area, Perimeter, and Volume

Square

A = s2

P = 4s

Cube

V = s3

s

s

s

ss

Rectangle

A = lwP = 2l + 2w

RectangularSolid

V = lwh

l

w

h

lw

Circle

A = pr2

C = 2pr

CircularCylinder

V = pr2h

r

r

Triangle

A = qbh

Sphere

V = dpr3

b

h

r

Trapezoid

A = qh(a + b)

Cone

V = apr2h

a

b

h

h

r

h

We will be using the formula for the perimeter of a rectangle, inour next example.The formula states that a rectangle’s perimeter is the sum of twiceits length and twice its width.

Finding the Dimensions of an American Football Field

The length of an American football field is 200 feet more than the width. If theperimeter of the field is 1040 feet, what are its dimensions?

SolutionStep 1 Let represent one of the unknown quantities. We know something aboutthe length; the length is 200 feet more than the width. We will let

x = the width.

x

EXAMPLE 6

P = 2l + 2w,

C-BLTZMC01_089-196-hr2 12-09-2008 10:39 Page 123


x + 200

Width

Length

x

Figure 1.17 An Americanfootball field

Step 2 Represent other unknown quantities in terms of Because the length is200 feet more than the width, we add 200 to the width to represent the length. Thus,

Figure 1.17 illustrates an American football field and its dimensions.

Step 3 Write an equation in that models the conditions. Because the perimeterof the field is 1040 feet,


This is the equation that models theproblem’s conditions.

Apply the distributive property.

Combine like terms:


Divide both sides by 4.

Thus,

The dimensions of an American football field are 160 feet by 360 feet. (The 360-footlength is usually described as 120 yards.)

Step 5 Check the proposed solution in the original wording of the problem. Theperimeter of the football field using the dimensions that we found is

Because the problem’s wording tells us that the perimeter is 1040 feet, ourdimensions are correct.

Check Point 6 The length of a rectangular basketball court is 44 feet more thanthe width. If the perimeter of the basketball court is 288 feet, what are itsdimensions?

Solving a Formula for One of Its VariablesWe know that solving an equation is the process of finding the number (or numbers)that make the equation a true statement. All of the equations we have solvedcontained only one letter,

By contrast, formulas contain two or more letters, representing two or morevariables. An example is the formula for the perimeter of a rectangle:

We say that this formula is solved for the variable because is alone on one sideof the equation and the other side does not contain a

Solving a formula for a variable means rewriting the formula so that thevariable is isolated on one side of the equation. It does not mean obtaining anumerical value for that variable.

To solve a formula for one of its variables, treat that variable as if it were the onlyvariable in the equation. Think of the other variables as if they were numbers. Isolateall terms with the specified variable on one side of the equation and all terms withoutthe specified variable on the other side. Then divide both sides by the same nonzeroquantity to get the specified variable alone.The next two examples show how to do this.

P.PP

P = 2l + 2w.

x.

21160 feet2 + 21360 feet2 = 320 feet + 720 feet = 1040 feet.

length = x + 200 = 160 + 200 = 360.

width = x = 160.

x = 160

4x = 640

2x + 2x = 4x. 4x + 400 = 1040

2x + 400 + 2x = 1040

2(x+200)+2x=1040

plus isTwice thelength

twice thewidth

the perimeter.

2(x+200) + 2x = 1040.

x

x + 200 = the length.

x.

� Solve a formula for a variable.

C-BLTZMC01_089-196-hr2 12-09-2008 10:39 Page 124


Solving a Formula for a Variable

Solve the formula for

Solution First, isolate on the right by subtracting from both sides. Thensolve for by dividing both sides by 2.

This is the given formula.

Isolate by subtracting fromboth sides.

Simplify.

Solve for by dividing both sides by 2.

Simplify.

Equivalently,

Check Point 7 Solve the formula for

Solving a Formula for a Variable That Occurs Twice

The formula

describes the amount, that a principal of dollars is worth after years wheninvested at a simple annual interest rate, Solve this formula for

Solution Notice that all the terms with already occur on the right side of theformula.

We can factor from the two terms on the right to convert the two occurrences ofinto one.

This is the given formula.

Factor out on the right side ofthe equation.

Divide both sides by

Simplify:

Equivalently,

Check Point 8 Solve the formula for C.P = C + MC

P =

A

1 + rt.

P11 + rt2

111 + rt2=

P1

= P. A

1 + rt= P

1 + rt. A

1 + rt=

P11 + rt2

1 + rt

P A = P11 + rt2

A = P + Prt

PP

We need to isolate P.

A=P+Prt

P

P.r.tPA,

A = P + Prt

EXAMPLE 8

w.P = 2l + 2w

l =

P - 2w

2.

P - 2w

2= l

l P - 2w

2=

2l

2

P - 2w = 2l

2w2l P - 2w = 2l + 2w - 2w

We need to isolate l.

P=2l+2w

l2w2l

l.P = 2l + 2w

EXAMPLE 7

Study TipYou cannot solve for

by subtracting from both sidesand writing

When a formula is solved for aspecified variable, that variable mustbe isolated on one side. The variable

occurs on both sides of

A - Prt = P.

P

A - Prt = P.

PrtPA = P + Prt

C-BLTZMC01_089-196-hr3 16-09-2008 11:48 Page 125


Exercise Set 1.3

Practice ExercisesUse the five-step strategy for solving word problems to find thenumber or numbers described in Exercises 1–10.

1. When five times a number is decreased by 4, the result is 26.What is the number?

2. When two times a number is decreased by 3, the result is 11.What is the number?

3. When a number is decreased by 20% of itself, the result is 20.What is the number?

4. When a number is decreased by 30% of itself, the result is 28.What is the number?

5. When 60% of a number is added to the number, the result is192. What is the number?

6. When 80% of a number is added to the number, the result is252. What is the number?

7. 70% of what number is 224?8. 70% of what number is 252?9. One number exceeds another by 26. The sum of the numbers

is 64. What are the numbers?10. One number exceeds another by 24. The sum of the numbers

is 58. What are the numbers?

Practice PlusIn Exercises 11–18, find all values of satisfying the given conditions.

11. and exceeds by 2.12. and exceeds by 3.13. and is 14 more than

8 times 14. and is 51 less than 12 times 15. and the difference between

3 times and 5 times is 22 less than 16. and the difference between

2 times and 3 times is 8 less than 4 times

17. and the sum of 3 times and

4 times is the product of 4 and

18. and the difference

between 6 times and 3 times is the product of 7 and

Application Exercises19. The bar graph shows the time Americans spent using various

media in 2007.

y3 .y2y1

y1 =

1x

, y2 =

1x2

- x, y3 =

1x - 1

,

y3 .y2

y1y1 =

1x

, y2 =

12x

, y3 =

1x - 1

,

y3 .y2y1

y1 = 2.5, y2 = 2x + 1, y3 = x,y3 .y2y1

y1 = 2x + 6, y2 = x + 8, y3 = x,y2 .y1y1 = 913x - 52, y2 = 3x - 1,

y2 .y1y1 = 1012x - 12, y2 = 2x + 1,

y2y1y1 = 10x + 6, y2 = 12x - 7,y2y1y1 = 13x - 4, y2 = 5x + 10,

x

Time spent watching TV exceeded time spent listening to theradio by 581 hours. The combined time devoted to these twomedia was 2529 hours. In 2007, how many hours didAmericans spend listening to the radio and how many hourswere spent watching TV?

20. Compared with Europeans, American employees use lessvacation time.

Average Number of Hours AmericansUsed Various Media in 2007

1800

600

900

1200

1500

175195 175300

Num

ber

of H

ours

Watching TV

Listening to radio

Using the Internet

Reading the newspaper

Listening to music

Source: Communication Industry Forecast and Report

Average Vacation Time forEuropeans and Americans

9

4

5

6

8

6.6

3

Vac

atio

n T

ime

(wee

ks)

77.0

7.8

Italy Germany France England U.S.

Source: The State of Working America 2006/2007

The average time Italians spend on vacation exceeds theaverage American vacation time by 4 weeks. The combinedaverage vacation time for Americans and Italians is11.8 weeks. On average, how many weeks do Americansspend on vacation and how many weeks do Italians spend onvacation?

Exercises 21–22 involve the average salaries represented by the bargraph.

Average Salaries for Various Jobs

$70,000

$20,000

$30,000

$40,000

$50,000

$60,000

15,080

$10,000

Ave

rage

Sal

ary Carpenters

Janitors Fast-food cooks

Computer programmers

Registered nurses

Source: 2007 data from salary.com

21. The average salary for computer programmers is $7740 lessthan twice the average salary for carpenters. Combined, theiraverage salaries are $99,000. Determine the average salaryfor each of these jobs.

22. The average salary for registered nurses is $3500 less thanthree times the average salary for janitors. Combined, theiraverage salaries are $74,060. Determine the average salaryfor each of these jobs.

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The bar graph indicates that American attitudes have becomemore tolerant over two decades on a variety of issues. Exercises23–24 are based on the data displayed by the graph.

28. Video Store A charges $9 to rent a video game for one week.Although only members can rent from the store, membershipis free. Video Store B charges only $4 to rent a video gamefor one week. Only members can rent from the store andmembership is $50 per year. After how many video-gamerentals will the total amount spent at each store be the same?What will be the total amount spent at each store?

29. The bus fare in a city is $1.25. People who use the bus havethe option of purchasing a monthly discount pass for $15.00.With the discount pass, the fare is reduced to $0.75.Determine the number of times in a month the bus must beused so that the total monthly cost without the discount passis the same as the total monthly cost with the discount pass.

30. A discount pass for a bridge costs $30 per month. The toll forthe bridge is normally $5.00, but it is reduced to $3.50 forpeople who have purchased the discount pass. Determine thenumber of times in a month the bridge must be crossed sothat the total monthly cost without the discount pass is thesame as the total monthly cost with the discount pass.

31. In 2005, there were 13,300 students at college A, with aprojected enrollment increase of 1000 students per year. Inthe same year, there were 26,800 students at college B, with aprojected enrollment decline of 500 students per year.a. According to these projections, when will the colleges

have the same enrollment? What will be the enrollmentin each college at that time?

b. Use the following table to numerically check your workin part (a).What equations were entered for and toobtain this table?

32. In 2000, the population of Greece was 10,600,000, withprojections of a population decrease of 28,000 people peryear. In the same year, the population of Belgium was10,200,000, with projections of a population decrease of12,000 people per year. (Source: United Nations) Accordingto these projections, when will the two countries have thesame population? What will be the population at that time?

33. After a 20% reduction, you purchase a television for $336.What was the television’s price before the reduction?

34. After a 30% reduction, you purchase a dictionary for $30.80.What was the dictionary’s price before the reduction?

35. Including 8% sales tax, an inn charges $162 per night. Findthe inn’s nightly cost before the tax is added.

36. Including 5% sales tax, an inn charges $252 per night. Findthe inn’s nightly cost before the tax is added.

Exercises 37–38 involve markup, the amount added to the dealer’scost of an item to arrive at the selling price of that item.

37. The selling price of a refrigerator is $584. If the markup is25% of the dealer’s cost, what is the dealer’s cost of therefrigerator?

38. The selling price of a scientific calculator is $15. If themarkup is 25% of the dealer’s cost, what is the dealer’s costof the calculator?

Y2Y1

50%

10%

30%

70%

90%

40%

20%

60%

80%

Per

cent

age

of U

.S. A

dult

sA

gree

ing

wit

h th

e St

atem

ent

1983

43%

2007

79%

1986

43%

2007

55%

1986

52%

2007

74%

Changing Attitudes in the United States

I approve ofmarriage betweenblacks and whites.

I approve of men andwomen living togetherwithout being married.

I approve of womenholding combat jobsin the armed forces.

Source: USA Today

23. In 1983, 43% of U.S. adults approved of marriage betweenblacks and whites. For the period from 1983 through 2007,the percentage approving of interracial marriage increasedon average by 1.5 each year. If this trend continues, by whichyear will all American adults approve of interracialmarriage?

24. In 1986, 43% of U.S. adults approved of men and womenliving together without being married. For the period from1986 through 2007, the percentage approving of cohabitationincreased on average by approximately 0.6 each year. If thistrend continues, by which year will 61% of all Americanadults approve of cohabitation?

25. A new car worth $24,000 is depreciating in value by $3000per year.

a. Write a formula that models the car’s value, in dollars,after years.

b. Use the formula from part (a) to determine after howmany years the car’s value will be $9000.

c. Graph the formula from part (a) in the first quadrant ofa rectangular coordinate system. Then show your solu-tion to part (b) on the graph.

26. A new car worth $45,000 is depreciating in value by $5000per year.

a. Write a formula that models the car’s value, in dollars,after years.

b. Use the formula from part (a) to determine after howmany years the car’s value will be $10,000.

c. Graph the formula from part (a) in the first quadrant ofa rectangular coordinate system. Then show your solu-tion to part (b) on the graph.

27. You are choosing between two health clubs. Club A offersmembership for a fee of $40 plus a monthly fee of $25. ClubB offers membership for a fee of $15 plus a monthly fee of$30. After how many months will the total cost at eachhealth club be the same? What will be the total cost foreach club?

xy,

xy,

C-BLTZMC01_089-196-hr2 12-09-2008 10:39 7


39. You invested $7000 in two accounts paying 6% and 8%annual interest. If the total interest earned for the year was$520, how much was invested at each rate?

40. You invested $11,000 in two accounts paying 5% and 8%annual interest. If the total interest earned for the year was$730, how much was invested at each rate?

41. Things did not go quite as planned. You invested $8000, partof it in stock that paid 12% annual interest. However, the restof the money suffered a 5% loss. If the total annual incomefrom both investments was $620, how much was invested ateach rate?

42. Things did not go quite as planned. You invested $12,000,part of it in stock that paid 14% annual interest. However,the rest of the money suffered a 6% loss. If the total annualincome from both investments was $680, how much wasinvested at each rate?

43. A rectangular soccer field is twice as long as it is wide. If theperimeter of the soccer field is 300 yards, what are itsdimensions?

44. A rectangular swimming pool is three times as long as it iswide. If the perimeter of the pool is 320 feet, what are itsdimensions?

45. The length of the rectangular tennis court at Wimbledon is6 feet longer than twice the width. If the court’s perimeter is228 feet, what are the court’s dimensions?

46. The length of a rectangular pool is 6 meters less than twicethe width. If the pool’s perimeter is 126 meters, what are itsdimensions?

47. The rectangular painting in the figure shown measures12 inches by 16 inches and is surrounded by a frame ofuniform width around the four edges. The perimeter of therectangle formed by the painting and its frame is 72 inches.Determine the width of the frame.

48. The rectangular swimming pool in the figure shownmeasures 40 feet by 60 feet and is surrounded by a path ofuniform width around the four edges. The perimeter of therectangle formed by the pool and the surrounding path is 248feet. Determine the width of the path.

60 feet

40 feet x

x

12 in.

x

x

16 in.

49. An automobile repair shop charged a customer $448, listing$63 for parts and the remainder for labor. If the cost of laboris $35 per hour, how many hours of labor did it take to repairthe car?

50. A repair bill on a sailboat came to $1603, including $532 forparts and the remainder for labor. If the cost of labor is $63per hour, how many hours of labor did it take to repair thesailboat?

51. An HMO pamphlet contains the following recommendedweight for women: “Give yourself 100 pounds for the first5 feet plus 5 pounds for every inch over 5 feet tall.” Using thisdescription, what height corresponds to a recommendedweight of 135 pounds?

52. A job pays an annual salary of $33,150, which includes aholiday bonus of $750. If paychecks are issued twice a month,what is the gross amount for each paycheck?

53. Answer the question in the following Peanuts cartoon strip.(Note: You may not use the answer given in the cartoon!)

54. The rate for a particular international person-to-persontelephone call is $0.43 for the first minute, $0.32 for eachadditional minute, and a $2.10 service charge. If the cost of acall is $5.73, how long did the person talk?

In Exercises 55–74, solve each formula for the specified variable.Do you recognize the formula? If so, what does it describe?

55. for 56. for

57. for 58. for

59. for 60. for

61. for 62. for

63. for 64. for

65. for 66. for

67. for 68. for

69. for 70. for

71. for 72.for

73. for 74. for R11R

=

1R1

+

1R2

f1p

+

1q

=

1f

hA = 2lw + 2lh + 2whIIR + Ir = E

rS =

C

1 - rSB =

F

S - V

tS = P + PrtrS = P + Prt

bA =12 h1a + b2aA =

12 h1a + b2

MP = C + MCpT = D + pm

hV = pr2 hmE = mc2

rC = 2prPI = Prt

BV =13 BhbA =

12 bh

RD = RTwA = lw

PEANUTS reprinted by permission of United FeatureSyndicate, Inc.

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Writing in Mathematics75. In your own words, describe a step-by-step approach for

solving algebraic word problems.

76. Write an original word problem that can be solved using alinear equation. Then solve the problem.

77. Explain what it means to solve a formula for a variable.

78. Did you have difficulties solving some of the problems thatwere assigned in this exercise set? Discuss what you did ifthis happened to you. Did your course of action enhanceyour ability to solve algebraic word problems?

Technology Exercises79. Use a graphing utility to numerically or graphically verify

your work in any one exercise from Exercises 27–30. Forassistance on how to do this, refer to the Technology box onpage 121.

80. A tennis club offers two payment options. Members can paya monthly fee of $30 plus $5 per hour for court rental time.The second option has no monthly fee, but court time costs$7.50 per hour.

a. Write a mathematical model representing total monthlycosts for each option for hours of court rental time.

b. Use a graphing utility to graph the two models in a [0, 15, 1]by [0, 120, 20] viewing rectangle.

c. Use your utility’s trace or intersection feature todetermine where the two graphs intersect. Describe whatthe coordinates of this intersection point represent inpractical terms.

d. Verify part (c) using an algebraic approach by setting thetwo models equal to one another and determining howmany hours one has to rent the court so that the twoplans result in identical monthly costs.

Critical Thinking ExercisesMake Sense? In Exercises 81–84, determine whether eachstatement makes sense or does not make sense, and explain yourreasoning.

81. By modeling attitudes of college freshmen from 1969through 2006, I can make precise predictions about theattitudes of the freshman class of 2020.

82. I find the hardest part in solving a word problem is writingthe equation that models the verbal conditions.

83. I solved the formula for one of its variables, so now I have anumerical value for that variable.

84. After a 35% reduction, a computer’s price is $780, so Idetermined the original price, by solving

85. At the north campus of a performing arts school, 10% of thestudents are music majors. At the south campus, 90% of thestudents are music majors. The campuses are merged intoone east campus. If 42% of the 1000 students at the eastcampus are music majors, how many students did the northand south campuses have before the merger?

86. The price of a dress is reduced by 40%. When the dress stilldoes not sell, it is reduced by 40% of the reduced price. If theprice of the dress after both reductions is $72, what was theoriginal price?

x - 0.35 = 780.x,

x

87. In a film, the actor Charles Coburn plays an elderly “uncle”character criticized for marrying a woman when he is 3 timesher age. He wittily replies, “Ah, but in 20 years time I shallonly be twice her age.” How old is the “uncle” and thewoman?

88. Suppose that we agree to pay you 8¢ for every problem inthis chapter that you solve correctly and fine you 5¢ for everyproblem done incorrectly. If at the end of 26 problems we donot owe each other any money, how many problems did yousolve correctly?

89. It was wartime when the Ricardos found out Mrs. Ricardowas pregnant. Ricky Ricardo was drafted and made out awill, deciding that $14,000 in a savings account was to bedivided between his wife and his child-to-be. Ratherstrangely, and certainly with gender bias, Ricky stipulatedthat if the child were a boy, he would get twice the amount ofthe mother’s portion. If it were a girl, the mother would gettwice the amount the girl was to receive. We’ll never knowwhat Ricky was thinking of, for (as fate would have it) he didnot return from war. Mrs. Ricardo gave birth to twins—a boyand a girl. How was the money divided?

90. A thief steals a number of rare plants from a nursery. On theway out, the thief meets three security guards, one afteranother. To each security guard, the thief is forced to giveone-half the plants that he still has, plus 2 more. Finally, thethief leaves the nursery with 1 lone palm. How many plantswere originally stolen?

91. Solve for

Group Exercise92. One of the best ways to learn how to solve a word problem in

algebra is to design word problems of your own. Creating aword problem makes you very aware of precisely how muchinformation is needed to solve the problem. You must alsofocus on the best way to present information to a reader andon how much information to give. As you write yourproblem, you gain skills that will help you solve problemscreated by others.

The group should design five different word problemsthat can be solved using linear equations.All of the problemsshould be on different topics. For example, the group shouldnot have more than one problem on simple interest. Thegroup should turn in both the problems and their algebraicsolutions.

(If you’re not sure where to begin, consider the graph forExercises 23–24 and the data that we did not use regardingattitudes about women in combat.)

Preview ExercisesExercises 93–95 will help you prepare for the material covered inthe next section.

93. Multiply:

94. Simplify:

95. Rationalize the denominator:7 + 422

2 - 522.

218 - 28.

17 - 3x21-2 - 5x2.

C: V = C -

C - S

L N.

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section 1.3 models and applications

Documents