Equations and Inequalities · Absolute Value 1.7 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications • Chapter Review • Chapter Test • Chapter

81

Equations andInequalities

Outline

Financing a PurchaseWhenever we make a major purchase, such as an automobile or a house,we often need to finance the purchase by borrowing money from alending institution, such as a bank. Have you ever wondered how thebank determines the monthly payment? How much total interest willbe paid over the course of the loan? What roles do the rate of interestand the length of the loan play?

—See the Internet-based Chapter Project I—

1.1 Linear Equations1.2 Quadratic Equations1.3 Complex Numbers; Quadratic Equations

in the Complex Number System1.4 Radical Equations; Equations Quadratic

in Form; Factorable Equations

1.5 Solving Inequalities1.6 Equations and Inequalities Involving

Absolute Value1.7 Problem Solving: Interest, Mixture,

Uniform Motion, Constant Rate JobApplications

• Chapter Review• Chapter Test• Chapter Projects

Chapter 1, Equations and Inequalities, reviews many topics covered in IntermediateAlgebra. If your instructor decides to exclude complex numbers from the course, don’t be alarmed. The book has beendesigned so that the topic of complex numbers can be included or excluded without any confusion later on.

134 CHAPTER 1 Equations and Inequalities

In Problems 71–76, find a and b.

71. If then

72. If then

73. If then

74. If then

75. If then

76. If then

77. Show that: if and , then

[Hint: .]

78. Show that

79. Prove the triangle inequality

[Hint: Expand and use the result ofProblem 78.]

ƒa + b ƒ2 = 1a + b22,ƒa + b ƒ … ƒa ƒ + ƒb ƒ .a … ƒa ƒ .

b - a = 11b - 1a211b + 1a2 a 6 b.1a 6 1ba 7 0, b 7 0,

a … 1x + 5

… b.ƒx + 1 ƒ … 3,

a … 1x - 10

… b.ƒx - 2 ƒ … 7,

a … 3x + 1 … b.ƒx - 3 ƒ … 1,

a … 2x - 3 … b.ƒx + 4 ƒ … 2,

a 6 x - 2 6 b.ƒx + 2 ƒ 6 5,

a 6 x + 4 6 b.ƒx - 1 ƒ 6 3,80. Prove that

[Hint: Apply the triangle inequality from Problem 79 to]

81. If show that the solution set of the inequality

consists of all numbers x for which

82. If show that the solution set of the inequality

consists of all numbers x for which

x 6 -1a or x 7 1ax2 7 a

a 7 0,

-1a 6 x 6 1ax2 6 a

a 7 0,

ƒa ƒ = ƒ1a - b2 + b ƒ .

ƒa - b ƒ Ú ƒa ƒ - ƒb ƒ .

93. The equation has no solution. Explain why.

94. The inequality has all real numbers as solutions.Explain why.

ƒx ƒ 7 -0.5ƒx ƒ = -2 95. The inequality has as solution set Explain

why.5x ƒx Z 06.ƒx ƒ 7 0

Explaining Concepts: Discussion and Writing

1. 2 2. True

‘Are You Prepared?’ Answers

1.7 Problem Solving: Interest, Mixture, Uniform Motion,Constant Rate Job Applications

OBJECTIVES 1 Translate Verbal Descriptions into Mathematical Expressions (p. 135)

2 Solve Interest Problems (p. 136)

3 Solve Mixture Problems (p. 137)

4 Solve Uniform Motion Problems (p. 138)

5 Solve Constant Rate Job Problems (p. 140)

Applied (word) problems do not come in the form “Solve the equation ”Instead, they supply information using words, a verbal description of the real problem.So, to solve applied problems, we must be able to translate the verbal descriptioninto the language of mathematics.We do this by using variables to represent unknownquantities and then finding relationships (such as equations) that involve thesevariables. The process of doing all this is called mathematical modeling.

. Á

In Problems 83–90, use the results found in Problems 81 and 82 to solve each inequality.

86. x2 Ú 1 87. x2 … 16 88. x2 … 9

89. x2 7 4 90. x2 7 16 91. Solve ƒ3x - ƒ2x + 1 ƒ ƒ = 4.

83. x2 6 1 84. x2 6 4 85. x2 Ú 9

92. Solve ƒx + ƒ3x - 2 ƒ ƒ = 2.

SECTION 1.7 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications 135

Real problem

Verbaldescription

Language ofmathematics

Mathematical problem

Solution

CheckCheckCheck

Figure 15

1 Translate Verbal Descriptions into Mathematical Expressions

Translating Verbal Descriptions into Mathematical Expressions

(a) For uniform motion, the constant speed of an object equals the distance traveleddivided by the time required.

Translation: If is the speed, d the distance, and t the time, then

(b) Let x denote a number.The number 5 times as large as x is The number 3 less than x is The number that exceeds x by 4 is The number that, when added to x, gives 5 is

Now Work P R O B L E M 7

Always check the units used to measure the variables of an applied problem. InExample 1(a), if is measured in miles per hour, then the distance d must beexpressed in miles and the time t must be expressed in hours. It is a good practice tocheck units to be sure that they are consistent and make sense.

The steps to follow for solving applied problems, given earlier, are repeatednext:

r

5 - x.x + 4.

x - 3.5x.

r = dt

.r

EXAMPLE 1

!

Steps for Solving Applied Problems

STEP 1: Read the problem carefully, perhaps two or three times. Pay particularattention to the question being asked in order to identify what you arelooking for. If you can, determine realistic possibilities for the answer.

STEP 2: Assign a letter (variable) to represent what you are looking for, and, ifnecessary, express any remaining unknown quantities in terms of thisvariable.

STEP 3: Make a list of all the known facts, and translate them into mathemati-cal expressions.These may take the form of an equation or an inequal-ity involving the variable. If possible, draw an appropriately labeleddiagram to assist you. Sometimes a table or chart helps.

STEP 4: Solve the equation for the variable, and then answer the question.STEP 5: Check the answer with the facts in the problem. If it agrees, congratu-

lations! If it does not agree, try again.

Any solution to the mathematical problem must be checked against the mathe-matical problem, the verbal description, and the real problem. See Figure 15 for anillustration of the modeling process.


2 Solve Interest ProblemsInterest is money paid for the use of money. The total amount borrowed (whetherby an individual from a bank in the form of a loan or by a bank from an individualin the form of a savings account) is called the principal. The rate of interest,expressed as a percent, is the amount charged for the use of the principal for a givenperiod of time, usually on a yearly (that is, on a per annum) basis.

Simple Interest Formula

If a principal of P dollars is borrowed for a period of t years at a per annuminterest rate r, expressed as a decimal, the interest I charged is

(1)I = Prt

Interest charged according to formula (1) is called simple interest. When usingformula (1), be sure to express r as a decimal.

Finance: Computing Interest on a Loan

Suppose that Juanita borrows $500 for 6 months at the simple interest rate of 9%per annum.What is the interest that Juanita will be charged on the loan? How muchdoes Juanita owe after 6 months?

EXAMPLE 2

Solution The rate of interest is given per annum, so the actual time that the money isborrowed must be expressed in years. The interest charged would be the principal,

$500, times the rate of interest times the time in years,

After 6 months, Juanita will owe what she borrowed plus the interest:

$500 + $22.50 = $522.50

Interest charged = I = Prt = 1500210.092a12b = $22.50

12

:19% = 0.092

!

Financial Planning

Candy has $70,000 to invest and wants an annual return of $2800, which requires anoverall rate of return of 4%. She can invest in a safe, government-insured certificateof deposit, but it only pays 2%. To obtain 4%, she agrees to invest some of hermoney in noninsured corporate bonds paying 7%. How much should be placed ineach investment to achieve her goal?

EXAMPLE 3

Solution STEP 1: The question is asking for two dollar amounts: the principal to invest in thecorporate bonds and the principal to invest in the certificate of deposit.

STEP 2: We let x represent the amount (in dollars) to be invested in the bonds.Thenis the amount that will be invested in the certificate. (Do you

see why?)STEP 3: We set up a table:

70,000 - x

Principal ($) Rate Time (yr) Interest ($)

Bonds x 1

Certificate 1

Total 70,000 1 0.04(70,000) = 28004% = 0.04

0.02(70,000 - x)2% = 0.0270,000 - x

0.07x7% = 0.07


Since the total interest from the investments is equal to we have the equation

(Note that the units are consistent: the unit is dollars on each side.)STEP 4:

Candy should place $28,000 in the bonds and in the certificate.

STEP 5: The interest on the bonds after 1 year is the intereston the certificate after 1 year is The total annualinterest is $2800, the required amount.

Now Work P R O B L E M 1 7

0.021$42,0002 = $840.0.071$28,0002 = $1960;

$42,000=$70,000 - $28,000 x = 28,000

0.05x = 1400 0.07x + 1400 - 0.02x = 2800

0.07x + 0.02170,000 - x2 = 2800

0.04170,0002 = 2800,

3 Solve Mixture ProblemsOil refineries sometimes produce gasoline that is a blend of two or more types offuel; bakeries occasionally blend two or more types of flour for their bread. Theseproblems are referred to as mixture problems because they combine two or morequantities to form a mixture.

Blending Coffees

The manager of a Starbucks store decides to experiment with a new blend of coffee.She will mix some B grade Colombian coffee that sells for $5 per pound with someA grade Arabica coffee that sells for $10 per pound to get 100 pounds of the newblend. The selling price of the new blend is to be $7 per pound, and there is to be nodifference in revenue from selling the new blend versus selling the other types. Howmany pounds of the B grade Colombian and A grade Arabica coffees are required?

EXAMPLE 4

Solution Let x represent the number of pounds of the B grade Colombian coffee. Thenequals the number of pounds of the A grade Arabica coffee. See Figure 16.100 - x

$5 per pound

!

B GradeColombianx pounds !

$10 per pound

"

A GradeArabica

100 − x pounds "

$7 per pound

Blend

100 pounds

Figure 16

Since there is to be no difference in revenue between selling the A and B gradesseparately versus the blend, we have

100#$7=1100 - x2#$10+x#$5

ePrice per poundof B grade f e# Pounds

B grade f + ePrice per poundof A grade f e# Pounds

A grade f = ePrice per poundof blend f e # Pounds

blend f

!

Distance mi

Time hr

Rate mi/hr


We have the equation

The manager should blend 60 pounds of B grade Colombian coffee withpounds of A grade Arabica coffee to get the desired blend.

Check: The 60 pounds of B grade coffee would sell for and the40 pounds of A grade coffee would sell for the totalrevenue, $700, equals the revenue obtained from selling the blend, as desired.


1$1021402 = $400;1$521602 = $300,

100 - 60 = 40

x = 60 -5x = -300

5x + 1000 - 10x = 700 5x + 101100 - x2 = 700

!

4 Solve Uniform Motion ProblemsObjects that move at a constant speed are said to be in uniform motion. When theaverage speed of an object is known, it can be interpreted as its constant speed. Forexample, a bicyclist traveling at an average speed of 25 miles per hour is in uniformmotion.

Uniform Motion Formula

If an object moves at an average speed (rate) the distance d covered in timet is given by the formula

(2)

That is, Distance = Rate # Time.

d = rt

r,

Physics: Uniform Motion

Tanya, who is a long-distance runner, runs at an average speed of 8 miles per hourTwo hours after Tanya leaves your house, you leave in your Honda and

follow the same route. If your average speed is how long will it be beforeyou catch up to Tanya? How far will each of you be from your home?

40 mi>hr,(mi>hr).

EXAMPLE 5

Solution Refer to Figure 17. We use t to represent the time (in hours) that it takes the Hondato catch up to Tanya. When this occurs, the total time elapsed for Tanya is hours.

t + 2

Time t

2 hrt ! 0

Time t

t ! 0

Figure 17

Set up the following table:

Tanya 8

Honda 40 t 40t

8(t + 2)t + 2


Since the distance traveled is the same, we are led to the following equation:

It will take the Honda hour to catch up to Tanya. Each will have gone 20 miles.

Check: In 2.5 hours,Tanya travels a distance of In hour, the

Honda travels a distance of a12b1402 = 20 miles.

12

12.52182 = 20 miles.

12

t = 12

hour

32t = 16 8t + 16 = 40t

81t + 22 = 40t

Physics: Uniform Motion

A motorboat heads upstream a distance of 24 miles on a river whose current isrunning at 3 miles per hour The trip up and back takes 6 hours.Assuming thatthe motorboat maintained a constant speed relative to the water, what was its speed?

(mi/hr).

EXAMPLE 6

Solution See Figure 18. We use to represent the constant speed of the motorboat relativeto the water. Then the true speed going upstream is and the truespeed going downstream is Since then

Set up a table.Time = DistanceRate

.

Distance = Rate * Time,r + 3 mi/hr.r - 3 mi/hr,

r

Rate mi/hr

Distance mi

Timehr

!Distance

Rate

Upstream 24

Downstream 2424

r + 3r + 3

24r - 3

r - 3

Since the total time up and back is 6 hours, we have

Add the quotients on the left.

Simplify.

Multiply both sides by .

Place in standard form.

Divide by 6.

Factor.

Apply the Zero-Product Property and solve.

We discard the solution so the speed of the motorboat relative to thewater is


9 mi/hr.r = -1 mi/hr,

r = 9 or r = -1

1r - 921r + 12 = 0

r2 - 8r - 9 = 0

6r2 - 48r - 54 = 0

r 2 - 9 48r = 61r2 - 92 48rr2 - 9

= 6

241r + 32 + 241r - 321r - 321r + 32 = 6

24r - 3

+ 24r + 3

= 6

24 miles

r ! 3 mi/hr

r " 3 mi/hr

Figure 18

!

!


5 Solve Constant Rate Job ProblemsThis section involves jobs that are performed at a constant rate. Our assumption is

that, if a job can be done in t units of time, then of the job is done in 1 unit of time.1t

Working Together to Do a Job

At 10 AM Danny is asked by his father to weed the garden. From past experience,Danny knows that this will take him 4 hours, working alone. His older brother, Mike,when it is his turn to do this job, requires 6 hours. Since Mike wants to go golfingwith Danny and has a reservation for 1 PM, he agrees to help Danny. Assuming nogain or loss of efficiency, when will they finish if they work together? Can they makethe golf date?

EXAMPLE 7

Solution Set up Table 2. In 1 hour, Danny does of the job, and in 1 hour, Mike does of the

job. Let t be the time (in hours) that it takes them to do the job together. In 1 hour,

then, of the job is completed. We reason as follows:

From Table 2,

Working together, the job can be done in hours, or 2 hours, 24 minutes. They

should make the golf date, since they will finish at 12:24 PM.


125

t = 125

5t = 12

512

= 1t

3

12+ 2

12= 1t

14

+ 16

= 1t

aPart done by Dannyin 1 hour b + aPart done by Mike

in 1 hour b = aPart done togetherin 1 hour b

1t

16

14

!

Hours to Do Job

Part of Job Done in 1 Hour

Danny 4

Mike 6

Together t1t

16

14

Table 2

Concepts and Vocabulary1. The process of using variables to represent unknown quanti-

ties and then finding relationships that involve these vari-ables is referred to as .

2. The money paid for the use of money is .

3. Objects that move at a constant speed are said to be in.

4. True or False The amount charged for the use of principalfor a given period of time is called the rate of interest.

5. True or False If an object moves at an average speed the distance d covered in time t is given by the formula

6. Suppose that you want to mix two coffees in order to obtain100 pounds of a blend. If x represents the number of poundsof coffee A, write an algebraic expression that representsthe number of pounds of coffee B.

d = rt.

r,

1.7 Assess Your Understanding


In Problems 7–16, translate each sentence into a mathematical equation. Be sure to identify the meaning of all symbols.

23. Business: Mixing Nuts A nut store normally sells cashewsfor $9.00 per pound and almonds for $3.50 per pound. But atthe end of the month the almonds had not sold well, so, inorder to sell 60 pounds of almonds, the manager decided tomix the 60 pounds of almonds with some cashews and sellthe mixture for $7.50 per pound. How many pounds ofcashews should be mixed with the almonds to ensure nochange in the profit?

24. Business: Mixing Candy A candy store sells boxes ofcandy containing caramels and cremes. Each box sells for$12.50 and holds 30 pieces of candy (all pieces are the samesize). If the caramels cost $0.25 to produce and the cremescost $0.45 to produce, how many of each should be in a boxto make a profit of $3?

25. Physics: Uniform Motion A motorboat can maintain aconstant speed of 16 miles per hour relative to the water.The boat makes a trip upstream to a certain point in 20 min-utes; the return trip takes 15 minutes. What is the speed ofthe current? See the figure.

Applications and Extensions

7. Geometry The area of a circle is the product of the numberand the square of the radius.

8. Geometry The circumference of a circle is the product ofthe number and twice the radius.

9. Geometry The area of a square is the square of the lengthof a side.

10. Geometry The perimeter of a square is four times thelength of a side.

11. Physics Force equals the product of mass and acceleration.

12. Physics Pressure is force per unit area.

13. Physics Work equals force times distance.

14. Physics Kinetic energy is one-half the product of the massand the square of the velocity.

15. Business The total variable cost of manufacturing x dish-washers is $150 per dishwasher times the number of dish-washers manufactured.

16. Business The total revenue derived from selling x dish-washers is $250 per dishwasher times the number of dish-washers sold.

17. Financial Planning Betsy, a recent retiree, requires $6000per year in extra income. She has $50,000 to invest and caninvest in B-rated bonds paying 15% per year or in a certifi-cate of deposit (CD) paying 7% per year. How much moneyshould be invested in each to realize exactly $6000 ininterest per year?

18. Financial Planning After 2 years, Betsy (see Problem 17)finds that she will now require $7000 per year. Assumingthat the remaining information is the same, how should themoney be reinvested?

19. Banking A bank loaned out $12,000, part of it at the rate of8% per year and the rest at the rate of 18% per year. If the in-terest received totaled $1000, how much was loaned at 8%?

20. Banking Wendy, a loan officer at a bank, has $1,000,000 tolend and is required to obtain an average return of 18% peryear. If she can lend at the rate of 19% or at the rate of 16%,how much can she lend at the 16% rate and still meet herrequirement?

21. Blending Teas The manager of a store that specializes inselling tea decides to experiment with a new blend. She willmix some Earl Grey tea that sells for $5 per pound with someOrange Pekoe tea that sells for $3 per pound to get 100 poundsof the new blend. The selling price of the new blend is to be$4.50 per pound, and there is to be no difference in revenuefrom selling the new blend versus selling the other types. Howmany pounds of the Earl Grey tea and Orange Pekoe tea arerequired?

22. Business: Blending Coffee A coffee manufacturer wantsto market a new blend of coffee that sells for $3.90 perpound by mixing two coffees that sell for $2.75 and $5 perpound, respectively. What amounts of each coffee should beblended to obtain the desired mixture?

[Hint: Assume that the total weight of the desired blend is100 pounds.]

p

p

26. Physics: Uniform Motion A motorboat heads upstream ona river that has a current of 3 miles per hour. The trip up-stream takes 5 hours, and the return trip takes 2.5 hours.Whatis the speed of the motorboat? (Assume that the motorboatmaintains a constant speed relative to the water.)

27. Physics: Uniform Motion A motorboat maintained aconstant speed of 15 miles per hour relative to the water ingoing 10 miles upstream and then returning. The total timefor the trip was 1.5 hours. Use this information to find thespeed of the current.

28. Physics: Uniform Motion Two cars enter the FloridaTurnpike at Commercial Boulevard at 8:00 AM, eachheading for Wildwood. One car’s average speed is 10 milesper hour more than the other’s. The faster car arrives at

Wildwood at 11:00 AM, hour before the other car. What

was the average speed of each car? How far did each travel?

29. Moving Walkways The speed of a moving walkway istypically about 2.5 feet per second. Walking on such amoving walkway, it takes Karen a total of 40 seconds totravel 50 feet with the movement of the walkway and thenback again against the movement of the walkway. What isKaren’s normal walking speed?Source: Answers.com

12

1 0

2 0

3 0

4 0

1 02 0

3 04 0

SOUTH

TEDB


30. Moving Walkways The Gare Montparnasse train stationin Paris has a high-speed version of a moving walkway. If hewalks while riding this moving walkway, Jean Claude cantravel 200 meters in 30 seconds less time than if he standsstill on the moving walkway. If Jean Claude walks at anormal rate of 1.5 meters per second, what is the speed ofthe Gare Montparnasse walkway?Source: Answers.com

31. Tennis A regulation doubles tennis court has an area of2808 square feet. If it is 6 feet longer than twice its width,determine the dimensions of the court.Source: United States Tennis Association

32. Laser Printers It takes an HP LaserJet 1300 laser printer10 minutes longer to complete a 600-page print job by itselfthan it takes an HP LaserJet 2420 to complete the same jobby itself. Together the two printers can complete the job in12 minutes. How long does it take each printer to completethe print job alone? What is the speed of each printer?Source: Hewlett-Packard

33. Working Together on a Job Trent can deliver his newspa-pers in 30 minutes. It takes Lois 20 minutes to do the sameroute. How long would it take them to deliver thenewspapers if they work together?

34. Working Together on a Job Patrice, by himself, can paintfour rooms in 10 hours. If he hires April to help, they can dothe same job together in 6 hours. If he lets April work alone,how long will it take her to paint four rooms?

35. Enclosing a Garden A gardener has 46 feet of fencing tobe used to enclose a rectangular garden that has a border2 feet wide surrounding it. See the figure.(a) If the length of the garden is to be twice its width, what

will be the dimensions of the garden?(b) What is the area of the garden?(c) If the length and width of the garden are to be the same,

what would be the dimensions of the garden?(d) What would be the area of the square garden?

38. Computing Business Expense Therese, an outside sales-person, uses her car for both business and pleasure. Lastyear, she traveled 30,000 miles, using 900 gallons of gasoline.Her car gets 40 miles per gallon on the highway and 25 inthe city. She can deduct all highway travel, but no city travel,on her taxes. How many miles should Therese be allowed asa business expense?

39. Mixing Water and Antifreeze How much water should beadded to 1 gallon of pure antifreeze to obtain a solution thatis 60% antifreeze?

40. Mixing Water and Antifreeze The cooling system of acertain foreign-made car has a capacity of 15 liters. If thesystem is filled with a mixture that is 40% antifreeze, howmuch of this mixture should be drained and replaced bypure antifreeze so that the system is filled with a solutionthat is 60% antifreeze?

41. Chemistry: Salt Solutions How much water must beevaporated from 32 ounces of a 4% salt solution to make a6% salt solution?

42. Chemistry: Salt Solutions How much water must beevaporated from 240 gallons of a 3% salt solution to producea 5% salt solution?

43. Purity of Gold The purity of gold is measured in karats,with pure gold being 24 karats. Other purities of gold areexpressed as proportional parts of pure gold. Thus, 18-karat

gold is or 75% pure gold; 12-karat gold is or 50%

pure gold; and so on. How much 12-karat gold should bemixed with pure gold to obtain 60 grams of 16-karat gold?

44. Chemistry: Sugar Molecules A sugar molecule has twiceas many atoms of hydrogen as it does oxygen and one moreatom of carbon than oxygen. If a sugar molecule has atotal of 45 atoms, how many are oxygen? How many arehydrogen?

45. Running a Race Mike can run the mile in 6 minutes, andDan can run the mile in 9 minutes. If Mike gives Dan a headstart of 1 minute, how far from the start will Mike pass Dan?How long does it take? See the figure.

1224

,1824

,

2 ft

2 ft

36. Construction A pond is enclosed by a wooden deck that is3 feet wide. The fence surrounding the deck is 100 feet long.(a) If the pond is square, what are its dimensions?(b) If the pond is rectangular and the length of the pond is

to be three times its width, what are its dimensions?(c) If the pond is circular, what is its diameter?(d) Which pond has the most area?

37. Football A tight end can run the 100-yard dash in 12 seconds.A defensive back can do it in 10 seconds. The tight endcatches a pass at his own 20-yard line with the defensiveback at the 15-yard line. (See the figure.) If no other playersare nearby, at what yard line will the defensive back catchup to the tight end?

[Hint: At time the defensive back is 5 yards behindthe tight end.]

t = 0,

Start

mi1–4

mi3–4

mi1–2

MikeDan

Chapter Review 143

CHAPTER REVIEWThings to KnowQuadratic formula (pp. 97 and 110)

If then

If there are no real solutions.

Discriminant (pp. 97 and 110)If there are two distinct real solutions.

If there is one repeated real solution.

If there are no real solutions, but there are two distinct complex solutions that are not real; the complex solutionsare conjugates of each other.

b2 - 4ac 6 0,

b2 - 4ac = 0,

b2 - 4ac 7 0,

b2 - 4ac 6 0,

x =-b ; 4b2 - 4ac

2aax2 + bx + c = 0, a Z 0,

46. Range of an Airplane An air rescue plane averages300 miles per hour in still air. It carries enough fuel for 5hours of flying time. If, upon takeoff, it encounters a headwind of how far can it fly and return safely?(Assume that the wind remains constant.)

47. Emptying Oil Tankers An oil tanker can be emptied bythe main pump in 4 hours.An auxiliary pump can empty thetanker in 9 hours. If the main pump is started at 9 AM, whenshould the auxiliary pump be started so that the tanker isemptied by noon?

48. Cement Mix A 20-pound bag of Economy brand cementmix contains 25% cement and 75% sand. How muchpure cement must be added to produce a cement mix that is40% cement?

49. Emptying a Tub A bathroom tub will fill in 15 minuteswith both faucets open and the stopper in place. With bothfaucets closed and the stopper removed, the tub will emptyin 20 minutes. How long will it take for the tub to fill if bothfaucets are open and the stopper is removed?

50. Using Two Pumps A 5-horsepower (hp) pump can empty apool in 5 hours.A smaller, 2-hp pump empties the same poolin 8 hours. The pumps are used together to begin emptyingthis pool.After two hours, the 2-hp pump breaks down. Howlong will it take the larger pump to empty the pool?

51. A Biathlon Suppose that you have entered an 87-milebiathlon that consists of a run and a bicycle race. During

30 mi/hr,

39 oz.

7 in.

55. Critical Thinking You are the manager of a clothing storeand have just purchased 100 dress shirts for $20.00 each.After 1 month of selling the shirts at the regular price, youplan to have a sale giving 40% off the original selling price.However, you still want to make a profit of $4 on each shirtat the sale price. What should you price the shirts at initiallyto ensure this? If, instead of 40% off at the sale, you give50% off, by how much is your profit reduced?

56. Critical Thinking Make up a word problem that requiressolving a linear equation as part of its solution. Exchange prob-lems with a friend. Write a critique of your friend’s problem.

57. Critical Thinking Without solving, explain what is wrongwith the following mixture problem: How many liters of25% ethanol should be added to 20 liters of 48% ethanol toobtain a solution of 58% ethanol? Now go through an alge-braic solution. What happens?

58. Computing Average Speed In going from Chicago toAtlanta, a car averages 45 miles per hour, and in going fromAtlanta to Miami, it averages 55 miles per hour. If Atlanta ishalfway between Chicago and Miami, what is the averagespeed from Chicago to Miami? Discuss an intuitive solu-tion. Write a paragraph defending your intuitive solution.Then solve the problem algebraically. Is your intuitive solu-tion the same as the algebraic one? If not, find the flaw.

59. Speed of a Plane On a recent flight from Phoenix to KansasCity, a distance of 919 nautical miles, the plane arrived20 minutes early. On leaving the aircraft, I asked the captain,“What was our tail wind?” He replied, “I don’t know, but ourground speed was 550 knots.” How can you determine ifenough information is provided to find the tail wind? If possi-ble, find the tail wind. (1 knot = 1 nautical mile per hour)

Explaining Concepts: Discussion and Writing

your run, your average speed is 6 miles per hour, and duringyour bicycle race, your average speed is 25 miles per hour.You finish the race in 5 hours. What is the distance of therun? What is the distance of the bicycle race?

52. Cyclists Two cyclists leave a city at the same time, onegoing east and the other going west. The westbound cyclistbikes 5 mph faster than the eastbound cyclist. After 6 hoursthey are 246 miles apart. How fast is each cyclist riding?

53. Comparing Olympic Heroes In the 1984 Olympics,C. Lewis of the United States won the gold medal in the 100-meter race with a time of 9.99 seconds. In the 1896Olympics, Thomas Burke, also of the United States, won thegold medal in the 100-meter race in 12.0 seconds. If they ranin the same race repeating their respective times, by howmany meters would Lewis beat Burke?

54. Constructing a Coffee Can A 39-ounce can of Hills Bros.®

coffee requires 188.5 square inches of aluminum. If itsheight is 7 inches, what is its radius? [Hint: The surface areaS of a right cylinder is where r is theradius and h is the height.]

S = 2pr2 + 2prh,

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Equations and Inequalities · Absolute Value 1.7 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications • Chapter Review • Chapter Test • Chapter