Answers to Odd-Numbered Problems · AN-6 Answers to Odd-Numbered Problems (f) 11. (a) (b)y x 115 (c) (d) Window: X min 0; X max 100 Y min 1; Y max 120 y x 0 0 20406080100 20 40 60

y

x

–6

–14

–2 1 4

4(4, 2)

(2, –2)

(0, –6)

(–2, –10)

–4, –14)

(3, 0)

2x – y = 6

Answers to Odd-Numbered Problems

x

y

–2

–6

–6 2

6

10

14

6

(4, 12)

(2, 8)

(0, 4)(–2, 0)

(–4, –4)

y = 2x + 4

(–2, 3)

4

(2, 1)

321–1

x

43

1

–1–2

y

–2

19. m � � A slope of � means thatfor every 2 unit change in x, ywill change (�1) unit.

12

12

21. m � 0 A slope of 0 means that regardless how x changes, yremains constant.

(2, –1)(–3, –1)

31–1–2

x

321

–2–3

y

(–1, 2)

(–1, –2)

3 41 2–2–3

x

3

4

21

–2–3

y23. The slope is not defined.

7.x 0 3 2 �2 4 �4

y �6 0 �2 �10 2 �14

CHAPTER 1 Linear Equations

Exercise 1.1 (p. 15)

1. A � (4, 2); B � (6, 2); C � (5, 3); D � (�2, 1); E � (�2, �3);F � (3, �2); G � (6, �2); H � (5, 0)

3. The set of points of the form (2, y), where y is a real number, isa vertical line passing through (2, 0) on the x-axis.

5.x 0 �2 2 �2 4 �4

y 4 0 8 0 12 �4

4

(2, 4)

(2, –1)

(2, –3)

(2, 1)(2, 0)

31–1–2–3

x

4321

–1–2–3

y

9. (a) Vertical line: x � 2(b) Horizontal line: y � �3

11. (a) Vertical line: x � �4(b) Horizontal line: y � 1

13. m � A slope of means that for every 2 unit change in x,y changes 1 unit.

15. m � �1 A slope of �1 means that for every 1 unit changein x, y changes by (�1) units.

17. m � 3 A slope of 3 means that for every 1 unit change in x, y will change 3 units.

12

12

54

(1, 0)

(2, 3)

32–1

x

y

1

321

–1–2–3

AN-2 Answers to Odd-Numbered Problems

25. 27. 59. slope: m � ; y-intercept: (0, �2)23

(1, 2)

(2, 4)

3 41 2–1–2

x

4

21

–2

y

rise

run

x

y(–2, 7)

(2, 4)

–3 –1–1

1

3

5

7

1 3

rise

run

29. 31.

(–1, 3)

31 2–1–2–3

x

4

21

–2–1

y

(0, 3)

3 41 2–1–2–3

x

4

21

–2–3

–1

y

3

33. x � 2y � 0 35. x � y � 2

37. 2x � y � �9 39. 2x � 3y � �1

41. x � 2y � �5 43. 2x � y � 3

45. 3x � y � �12 47. 4x � 5y � 0

49. x � 2y � 2 51. x � 1

53. y � 4

55. slope: m � 2; y-intercept: (0, 3)

3 41 2–1–2–3

x

4

1

–2–3

–1

y

3

y = 2x + 3

(0, 3)

y

x

y = 2x – 2

–3 –1–1

1

3

–5

–7

1 3

(0, –2)

57. slope: m � 2; y-intercept:(0, �2)

3 41–1–2–3

x12

–1–2

–3

y

34

y = x – 223

(0, –2)

32 41–1–2

x

2

–1–2

y

3

y = –x + 1

(0, 1)

1–1–2–3–5

x1

–1–2–3–4–5

y

x = –4

(–4, 0)

31 2–1–2–3

x

4

6

23

1

y

y = 5

(0, 5)

61. slope: m � �1; y-intercept: (0, 1)

63. Slope is not defined; there is no y-intercept.



31 2–1–2–3

x

23

1

–2–3

–1

y y = x

(0, 0)

69. slope: m � ; y-intercept: (0, 0)32

31 2–1–2–3

x

23

1

–2–3

y y = x32

(0, 0)

71. y � �3 73. C � 0.122x

Exercise 1.2 AN-3

75. (a) C � 0.08275x � 7.58, 0 � x � 400(b)

(c) The monthly charge for using 100 KWH is $15.86.(d) The monthly charge for using 300 KWH is $32.41.(e) The slope indicates that for every extra KWH used (up

to 400 KWH), the electric bill increases by 8.275 cents.

77. w � 4h � 129 79. C � 0.53x � 1,070,000

81. (a) C � (F � 32) (b) C � 20°

83. (a) y �� t � 53.007 (b) y � 52.74 billion gallons(c) The slope tells us that the reservoir loses 1 billion gallons

of water every 75 days.(d) y � 52.594 billion gallons(e) In 10.89 years the reservoir will be empty.

85. Window: X min � �10; X max � 10Y min � �10; Y max � 10

175

59

x

y

00 200 400

20

40

(100, $15.86)

(300, $32.41)

–10

–10

10

10

–10

–10

10

10

x-intercept: (1.67, 0); y-intercept: (0, 2.50)


x-intercept: (2.52, 0); y-intercept: (0, �3.53)


–10

–10

10

10

–10

–10

10

10

91. Window: X min � �10; X max � 10;Y min � �10; Y max � 10

x-intercept: (0.78, 0); y-intercept: (0, �1.41)

93. (b)

95. (d)

97. y � x � 2 or x � y � �2

99. y � � x � 1 or x � 3y � 3

101. (b), (c), (e), (g)

103. y � 0

105. Answers vary.

107. No; no.

109. The lines are identical.

111. Two lines can have the same y-intercept and the same x-intercept but different slopes only if their y-intercept isthe point (0, 0).


1. parallel 3. intersecting 5. coincident

7. parallel 9. intersecting 11. intersecting

13

13. (3, 2) 15. (3, 1)

y

x(3, 2)

–3 –1

3

7

–5

–9

1 5

L M

(3, 1)

x

y

L

M

–5 –2

–3

1

5

1 4

x-intercept: (2.83, 0); y-intercept: (0, 2.56)


17. (1, 0) 19. (2, 1)

21. (�1, 1) 23. (4, �2)

43

(1, 0)

2–1

x

4

21

–1–2

yL M

43

(2, 1)

21–1–2

x

3

1

–2–3

y LM

3

(–1, 1)

21–1–3

x

3

1

–2–3

y LM

3 5

(4, –2)

21–1–2

x

3

12

–1

–3

y L

M

25. m1m2 � � (�3) � �1 27. m1m2 � � � (2) � �1

29. m1m2 � � (4) � �1 31. y � 2x � 3 or 2x � y � 3

33. y � � x � or x � 2y � 9

35. y � 4x � 6 or 4x � y � �6

37. y � 2x or 2x � y � 0 39. x � 4

41. y � � x � or x � 2y � �5

43. y � � x � or 15x � 30y � 19

45. t � 4 47. y � x � 49. (c)


1. (a) S � $80,000(b) S � $95,000(c) S � $105,000(d) S � $120,000

3. (a) y � 650x � 1,287,500(b) The average cost of a compact car is predicted to be

$15,750.(c) The slope can be interpreted as the average yearly

increase in price of a compact car.

5. (a) S � t �

(b) The predicted average SAT score will be 492.

7. (a) P � 0.53t � 1034.4(b) 27.7%(c) The slope is the annual percentage increase of the popu-

lation over 25 who hold bachelor’s degrees or higher.

16,5947

107

8519

519

1930

12

52

12

92

12

14

12

13

6050403020

(30, 900)

R

C

100

10001200

800600400200

0

y

xLoss

Profit

x

y

20000

50

100

150

200

400 600 800

RC

(500, 150)

Profit

Loss

9. The break even point is x � 30.

13. The break even point occurs when x � 1200 items are pro-duced and sold.

2000

1500

1000

500

00 1000 2000

x

y

(1200, 1200)

Loss

Profit

R

C

15. (a) The revenue from delivering x newspapers is given by R � $1.79x

(b) The cost of delivering x newspapers is given by C � $0.53x � $1,070,000

(c) The profit from delivering x newspapers is given byP � 1.26x � 1,070,000

(d) (849207, 1520080) The company breaks even when849,207 newspapers are delivered.

(e)

(f)

yx

1,000,000

300,000

0–200,000

–700,000

–1,200,000

P(849206, 0)

y

x

R

C2,000,000

1,500,000

1,000,000

500,000

00 1,000,000

(849207, 1520080)

11. The break even point is x � 500.

Exercise 1.4 AN-5

17. To break even put 20 caramels and 30 creams into each box;increase the number of caramels to obtain a profit.

19. Mr. Nicholson should invest $50,000 in AA Bonds and$100,000 in Savings and Loan Certificates.

21. Mix 25 pounds of Kona coffee with 75 pounds of Colombiancoffee to obtain a blend worth $10.80 per pound.

23. Mix 30 cubic centimeters of the 15% solution with 70 cubiccentimeters of the 5% solution to obtain a solution that is8% acid.

25. The market price is $1.00.

27. The market price is $10.00.

29. (a) Market price: $1.00(b) Supply demanded at market price: 1.1 units(c)

31. D � �4p � 23


1. A relation exists, and it appears to be linear.

3. A relation exists, and it appears to be linear.

5. No relation exists.

7. (a) (b) y � 2x � 2

(c)

x

y

–2

4

8

12

16

20

2 4 6 8 10

p

y

(1, 1.1)

S

D

3

2

1

00 1 2 3

(d) Window: X min � �2; X max � 10Y min � �3; Y max � 20

x

y

–2

4

8

12

16

20

2 4 6 8 10

–2

–3

20

10

(e) Using the LinReg program, the line of best fit is:y � 2.0357x � 2.3571.

(f)

–2

–3

20

10

9. (a)

(b) y � x �

(c)

(d) Window: X min � �6; X max � 6Y min � �6; Y max � 7

x

y

–6

–6

2

6

–2 2 6

12

94

x

y

–6

–6

2

6

–2 2 6

–6

–6

7

6

(e) Using the LinReg program, the line of best fit is:y � 2.2x � 1.2.


(f)

11. (a) (b) y � � x � 115

(c)

(d) Window: X min � 0; X max � 100Y min � 1; Y max � 120

y

x0

0 20 40 60 80 100

20

40

60

80

100

34y

x0

0 20 40 60 80 100

20

40

60

80

100

–6

–6

7

6

(b) y � 4x � 180(c)

(d) Window: X min � �30; X max � 10Y min � 0; Y max � 160

x

y

0

150

100

50

10–10–30 30

01

120

100

(e) Using the LinReg function the line of best fit is:y � �0.72x � 116.6.

(f)

13. (a)

x

y

0

150

100

50

10–10–30 30

01

120

100

–300

160

10

(e) Using the LinReg function, the line of best fit is:y � 3.86131x � 180.29197.

(f)

15. (a)

(b) C � I �

(c) The slope of this line indicates that a family will spend$23 of every extra $30 of disposable income.

(d) A family with a disposable income of $42,000 is predictedto consume $32,867 worth of goods.

(e) Using the LinReg function on the graphing utility, theline of best fit is: y � 0.75489x � 0.62663

17. (a) Window: X min � 0; X max � 75.5 (thousand)Y min � 0; Y max � 236.5 (thousand)

23

2330

00 20,000 40,000 60,000

20,000

40,000

60,000C

l

–300

160

10

00

236.5

75.5

Chapter 1 Review Exercises AN-7

(b) Using the LinReg function, the line of best fit is:y � 2.98140x � 0.07611.

(c)

(d) The slope indicates that a person can borrow an addi-tional $2.98 for each additional dollar of income.

(e) $125,143

19. (a) Window: X min � �10; X max � 110Y min � 50; Y max � 70

00

236.5

75.5

(b) Using the LinReg function, the line of best fit is:y � 0.07818x � 59.0909.

(c)

(d) The slope indicates the apparent change in temperaturein a 65°F room for every percent increase in relativehumidity.

(e) The apparent temperature of a room with an actual tem-perature of 65°F remains at 658. when the relative humi-tity is 75%.

–1050

70

110

–1050

70

110

CHAPTER 1 Review

True–False Items (p. 43)

1. F 2. T 3. T 4. F

5. F 6. T 7. F 8. T

9. F 10. F

Fill in the Blanks (p. 43)

1. abscissa; ordinate, or x-coordinate; y-coordinate

2. Undefined; zero 3. negative

4. parallel 5. coincident

6. perpendicular 7. intersecting

Review Exercises (p. 44)

1.

x

y

–4 –2

–2

2

–4

42

(0, 3)

(2, –1)

y = –2x + 3

3.

5. (a) m � � ; A slope � � means that for every 2 units x moves to the right, y moves down 1 unit.

(b) y � � x � or x � 2y � 5

(c)

7. (a) m � 2; A slope � 2 means that for every 1 unit changein x, y changes 2 units.

(b) y � 2x � 7 or 2x � y � �7(c) y

x

P

Q

–2–2 2 4 6

2

4

72x – y = –7

y

xP

Q

–3–2

2

4

5

x + 2y = 5

52

12

12

12

2 31–1

2y = 3x + 6

–3

x

21

y

3(0, 3)

(–2, 0)


9. y � �3x � 5 or 3x � y � 5

11. y � 4

13. x � 8

15. y � � x � 5 or 5x � 2y � 10

17. y � � x � 4 or 4x � 3y � �12

19. y � � or 2x � 3y � �1

y

x

(–5, 3)

(–2, 1)run

rise

–5 –2 3

3

–3

2x + 3y = –1

23 x �1

3

x

y

–3

–2

2

–4

3

(–3, 0)

(0, –4)

4x + 3y = –12

43

x

y

–4 –2–2

2

4

–4

4

(0, 5)

(2, 0)

(4, –5)

y = x + 552

52

–4

–4

4

8

4

y

x

(8, 5)

x = 8

x

y

–2

–4 –2

2

2 4

(–3, 4)

y = 4

y

x

rise

run

(1, 2)

(2, –1)

(3, –4)

3x + y = 5

–4

–4 –2–2

2

4

21. y � x � or 3x � 2y � �21

23. m � � , y-intercept (0, 9)

25. m � �2, y-intercept: (0, )

27. parallel 29. intersecting 31. coincident

33. (5, 1)

35. (1, 3) y

x(1, 3)

–4 4

–4

8M L

y

x(5, 1)

L

M–4

–4

–8

8

4 8

321–1–2–3

x

6

4

2

–2

y

4x + 2y = 9

92(0, )

94( , 0)

92

5439x + 2y = 18

1

x

810

642

y

2

(0, 9)

(2, 0)

92

3x – 2y = –21

y

x(–5, 3)

–4–4

4

12

–8

4–12 8

212(0, )

212

32


37. (�2, 1)

39. Invest $78,571.43 in B-rated bonds and $11,428.57 in thewell-known bank.

41. (a) 120 people need to attend for the group to break even.(b) 300 people need to attend to achieve the $900 profit.(c) If tickets are sold for $12.00 each 86 people must attend

to break even, and 215 people must attend to achieve aprofit of $900.

43. Relation does not appear to be linear.(a)

45. (a)

(b) m � �0.2733

8

6

4

2

1987 1989 1991 1993

Car

bon

mon

oxid

e

Year

90

45

1 3 5 7

y

x

(–2, 1)

–1–2–3–4

x

L

M

34

1

–1–2

y

2

(c) The slope indicates the average annual decrease in con-centration of carbon monoxide between 1987 and 1990.

(d) m � �0.33(e) The slope indicates the average annual decrease in con-

centration of carbon monoxide between 1990 and 1993.(f) m � �0.308(g) The slope indicates the average annual decrease in the

concentration of carbon monoxide.(h) The trend indicates that the average level of carbon

monoxide is decreasing.

47. (a)

(b) The relation appears to be linear.(c) m � 22.05(d) The slope represents the average annual increase in

value of a share of the Vanguard 500 Index Fund from1996 to 1999.

(e) y � 22.236x � 46.54 (f) $179.96

Mathematical Questions from Professional Exams

1. b 2. d 3. d 4. d

5. c 6. c 7. b 8. b

150

100

50

1996 1997 1998 1999Year

Val

ue

Exercise 2.1 AN-9

37. (�2, 1)

39. Invest $78,571.43 in B-rated bonds and $11,428.57 in thewell-known bank.

41. (a) 120 people need to attend for the group to break even.(b) 300 people need to attend to achieve the $900 profit.(c) If tickets are sold for $12.00 each 86 people must attend

to break even, and 215 people must attend to achieve aprofit of $900.

43. Relation does not appear to be linear.(a)

45. (a)

(b) m � �0.2733

8

6

4

2

1987 1989 1991 1993

Car

bon

mon

oxid

e

Year

90

45

1 3 5 7

y

x

(–2, 1)

–1–2–3–4

x

L

M

34

1

–1–2

y

2

(c) The slope indicates the average annual decrease in con-centration of carbon monoxide between 1987 and 1990.

(d) m � �0.33(e) The slope indicates the average annual decrease in con-

centration of carbon monoxide between 1990 and 1993.(f) m � �0.308(g) The slope indicates the average annual decrease in the

concentration of carbon monoxide.(h) The trend indicates that the average level of carbon

monoxide is decreasing.

47. (a)

(b) The relation appears to be linear.(c) m � 22.05(d) The slope represents the average annual increase in

value of a share of the Vanguard 500 Index Fund from1996 to 1999.

(e) y � 22.236x � 46.54 (f) $179.96


1. b 2. d 3. d 4. d

5. c 6. c 7. b 8. b

150

100

50

1996 1997 1998 1999Year

Val

ue


1. Yes 3. No 5. Yes 7. Yes 9. Yes

11. The solution of the system is x � 6 and y � 2.


15. The solution of the system is x � 8 and y � �4.

17. The solution of the system is x � and y � � .

19. The system is inconsistent.


16

13

CHAPTER 2 Systems of Linear Equations; Matrices.

23. The solutions of the system are y � � x � 2 and x where xis any real number, or as x � �2y � 4 and y, where y is anyreal number.


27. The solution of the system is x � and y � 1.


31. The solution of the system is x � and y � .

33. The solution of the system is x � 8, y � 2, and z � 0.

35. The solution of the system is x � 2, y � �1, and z � 1.

15

43

32

12



39. The solutions of the system are x � 5z � 2, y � 4z � 3, andz, where z is any real number.


43. The solution of the system is x � 1, y � 3, and z � �2.

45. The solution of the system is x � �3, y � , and z � 1.

47. The dimensions of the floor should be 30 ft � 15 ft.

49. They should plant 219.8 acres of corn and 225.2 acres ofsoybeans.

51. 22.5 pounds of cashews should be mixed with the peanuts.

53. A bowl of noodles costs 571 yen and a carton of milk costs220 yen.

55. The amount of the refund should be $5.56.

57. 50 mg of the first liquid (20% vitamin C and 30% vitaminD) should be mixed with 75 mg of the second liquid.

59. Use 9.16 pounds of rolled oats and 8.73 pounds of molasses.

61. The theater has 100 orchestra seats, 210 main seats, and 190balcony seats.

63. Kelly should invest $8000 in treasury bills, $7000 in treasurybonds, and $5000 in corporate bonds.


1. 3.

5. 7.

9.

11.

13.

15. (a) (b)

17. (a) (b)

19. (a) (b) �120

�3�5�8

167

� �2�2

0�� 10

�3

�311

144

� �226�

�120

�3�5

�15

238

� �6�4

�12�� 10

�3

�31

�6

2�1

2 � �6

86�

�120

�35

�6

46

16 � 3

615�� 1

0�3

�313

4�2

4 � 30

6��1

0�3

1 � �2

9��1

2�1

31

�1�1

4 � 05�

�410

�112

20

�1

�101

� 4�6

5��2

12

�312

1�1�3

� 71

� 4��213

�1�1�1

�110

� 012�

�23

11

� �6�1��2

1�3�1

� 53�

12

21.

The system is consistent and the solution is x � 7 and y � �1.

23.

The system is inconsistent.

25.

The system is consistent and has an infinite number of solu-tions. The solutions are x � �2z � �1, y � 4z � 2, and z,where z is any real number.

27.

The system is consistent. The solutions are x1 � �17x4 � 24,x2 � 7x4 � 10, x3 � �2x4 � 3, and x4, where x4 is any realnumber.

29.

The system is consistent. The solutions are x1 � 2x3 � 2x4 �4, x2 � �x3 � 3x4 � 3, x3 and x4, where x3 and x4 are anyreal numbers.

31.

The system is consistent. The solutions are x1 � x4 � 2,x2 � x4 � 2, x3 � x4, and x4, where x4 is any real number.






43. The solutions of the system are y � � x � and x, where x is any real number, or x � �3y � 2 and y where y is anyreal number.



49. The solution of the system is x � 1, y � 4, and z � 0.

51. The solution of the system is x � , y � � , and z � 6.



125

85

13

23

23

13

13

12

�x 1 � 2x 2 � x 4 � �2x 2 � 3x 3 � 2x 4 � 2

x 3 � x 4 � 00 � 0

�x 1 � 2x 2 � 4x 4 � 2x 2 � x 3 � 3x 4 � 3

0 � 0

�x 1 � 2x 2 � x 3 � x 4 � 1x 2 � 4x 3 � x 4 � 2

x 3 � 2x 4 � 3

�x � 2z � �1y � 4z � �2

0 � 0

�x � 2y � 3z � 1y � 4z � 2

0 � 3

�x � 2y � 5y � �1

Exercise 2.3 AN-11

57. The solution of the system is x � , y � , and z � 1.

59. The solution of the system is x � , y � � , and z � .


63. The solution of the system is x1 � 20, x2 � �13, x3 � , andx4 � .

65. A mezzanine ticket costs $54, a lower balcony ticket costs$38, and a middle balcony ticket costs $30.

67. There are 10 work stations set up for 2 students and 6 workstations set up for 3 students.

69. Carletta should invest $4000 in treasury bills, $4000 in trea-sury bonds, and $2000 in corporate bonds.

71. The meal should consist of 1 serving of chicken, 1 serving ofpotatoes, and 2.5 servings of spinach.

73. 20 cases of orange juice, 12 cases of tomato juice, and 6 casesof pineapple juice are prepared.

75. The teacher should order 2 of the first package (20 white, 15blue, 1 red), 10 of the second package (3 blue, 1 red), and 4of the third package.

77. The recreation center should purchase 4 assorted cartons, 8mixed cartons, and 5 single cartons.

79. To fill the order use 2 large cans, 1 mammoth can, and 4giant cans.

83. Answers will vary.


1. No, the second row contains all zeros; it should be at thebottom.

3. No, there is a 1 above the leftmost 1 in the 2nd row. It shouldbe a 0.

5. No, the leftmost 1 in the 2nd row is not to the right of theleftmost 1 in the 1st row.

7. Yes 9. Yes 11. Yes

13. Infinitely many solutions, x � �y � 1 and y, where y is anyreal number.

15. One solution, x � 4 and y � 5.

17. Infinitely many solutions, x � 2z � 6, and y � �3z � 1,and z, where z is any real number.

19. Infinitely many solutions, x � �2y � 1, y, and z � 2, wherey is any real number.

21. Infinitely many solutions, x � 1, y � 2, and z, where z, is anyreal number.

23. One solution, x � �1, y � 3, and z � 4.

25. Infinitely many solutions, x � z � 1, y � �2z � 1, and z,where z is any real number.

252

12

29

23

29

23

13 27. Infinitely many solutions, x1 � x4 � 4, and x2 � �2x3 �

3x4, x3, and x4, where x3 and x4 are any real numbers.



33. The solutions of the system are x � 2y � 4,

where y is any real number.

35. The solutions of the system are x � �3

y � �3z � 5, and z, where z is any real number.

37. The solution of the system is x � 3, y � 2,

and z � �4.

39. The solution of the system is

x1 � �17, x2 � 24, x3 � 33, and x4 � 14.

41. The solution of the system is

x1 � x4 � , x2 � x4 � , x3 � � x4 � , and x4, where x4 is any real number.



47. The solution of the system is x1 � 1,

x2 � 2, x3 � 0, and x4 � 1.

�1000

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0010

0001

� 1201�

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15

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51. The solutions of the system are x � 0,

y � z � 6, and z, where z is any real number.

53.

Amount in EE/E Amount in I Amount in HH/H

$ 0 $17,000 $8,000

$ 3,125 $14,875 $7,000

$ 6,250 $12,750 $6,000

$ 9,375 $10,625 $5,000

$12,500 $ 8,500 $4,000

$15,625 $ 6,375 $3,000

$18,750 $ 4,250 $2,000

$21,875 $ 2,125 $1,000

$25,000 $ 0 $ 0

55. Yes, the couple can still maintain their goals.

Amount in EE/E Amount in I Amount in HH/H

$ 0 $6,250 $18,750

$2,000 $4,550 $18,450

$4,000 $2,850 $18,150

$6,000 $1,150 $17,850

$7,353 $ 0 $17,647

57. There is insufficient information to determine the price ofeach food item.

Hamburger Large Fries Large Cola

$2.15 $0.88 $0.60

$2.10 $0.90 $0.65

$2.05 $0.92 $0.70

$2.00 $0.93 $0.75

$1.95 $0.95 $0.80

$1.90 $0.97 $0.85

$1.85 $0.98 $0.90

�100

010

0�1

0 � 0

�60�

�100

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� 001�

59. (a) The couple invests $20,000.

Treasury Bills Corporate Bonds Junk Bonds

$ 0 $10,000 $10,000

$1,000 $ 8,000 $11,000

$2,000 $ 6,000 $12,000

$3,000 $ 4,000 $13,000

$4,000 $ 2,000 $14,000

$5,000 $ 0 $15,000

(b) The couple invests $25,000.


$12,500 $12,500 $ 0

$13,500 $10,500 $1,000

$14,500 $ 8,500 $2,000

$15,500 $ 6,500 $3,000

$16,500 $ 4,500 $4,000

$17,500 $ 2,500 $5,000

$18,500 $ 500 $6,000

$18,750 $ 0 $6,250

(c) Even if all $30,000 are invested in treasury bills (the invest-ment with the lowest return), the interest income is $2100.

61.

Number of Number of Number of mg of 20% C, mg of 40% C, mg of 30% C, 30% D liquid 20% D liquid 50% D liquid

50 75 0

41.25 75.625 5

32.5 76.25 10

23.75 76.875 15

15 77.5 20

6.25 78.125 25

0 78.571 28.571



1. 2 � 2, a square matrix

3. 2 � 3 5. 3 � 2 7. 3 � 2

9. 2 � 1, a column matrix

Exercise 2.5 AN-13

11. 1 � 1, a column matrix, a row matrix, and a square matrix

13. False; the dimensions must be the same for two matrices tobe equal.

15. True 17. True 19. True 21. True

23. False; the dimensions must be the same for two matrices tobe equal.

25. 27.

29. 31.

33. 35.

37. 39.

41. 43.

45.

47.

49.

51. x � �4 and z � 4

53. x � 5 and y � 1

55. x � 4, y � �6, and z � 6

57.

59.

61.

63.

65. ,

Yes, a 4 � 2 matrix could represent the situation.

MaleFemale

�Associates

218,000364,000

Bachelor ,s573,000714,000

Master , s181,000261,000

Doctoral26,70020,400�

�Local

542,97870,556

State1,154,869

81,607

Fed135,237

10,179�MaleFemale

�37

420.529

�179

1618

3013

�322

151

3143�

�19

�1215.5

�29

�11�13�24

10

�14�21

19�14

�15�17�39�11

��

�24

�3.512

168

�1

77

�47

55

136�

2B � 3B � � 525

�105

010� � 5B

A � (�A) � �00

00

00� � 0

A � B � �35

�53

43� � B � A

� 23�17

�7�1

�18�17�� 5

�12�2

13

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28�17

143223��9

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1�6�3�

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18�6

03��1

617�

67.

69.


1. [14] 3. [4] 5. [18 �8]

7. 9. [4 6] 11.

13. 15.

17. BA is defined and is a 3 � 4 matrix.

19. AB is not defined.

21. (BA)C is not defined.

23. BA � A is defined and is a 3 � 4 matrix.

25. DC � B is defined and is a 3 � 3 matrix.

27. 29.

31.

33. 35.

37. 39.

41.

43.

45.

47. �6671

165158

7413

124.5�3

94106

79152

38285246�

�31.5

861369.5412.5

251350115882

�31.5791206.5451.5

143420215491

��

0.52119.5

�9.5

1614

845

�3028

�23�9

25643383�

D(CB) � ��61

�7

4220

4

�96

�18� � (DC)B

� 1015

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301620

3750

�8��14�20

7�6�

�84

432

2216��3

4�1

2�

� 68

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11

147��4

29

288�

� 412

�14�32��4

228�

�LAS250250

ENG225

75

EDUC80

120 �MaleFemale

�Democrats

351203

Republicans271215

Independents7355 �Under $25,000

Over $25,000


49.

51.

53. x � 1 or x � 55. a � d and b � �c

57.

59. Lee spent $372 while Chan spent $257.

61.

63.

67.

A10 �

433.097 �1583.562 �369.817 �216.548 �141.638

1207.695 5998.438 2563.227 �603.847 1045.705

�1423.023 8065.070 4271.798 711.511 2023.364

479.476 �246.538 �231.543 �239.737 �140.557

�899.341 �3064.313 �1627.502 449.670 �698.609

A15 �

�2247.845 �118449.318 �69122.364 1123.922 �32134.304

�11094.809 542433.513 249928.402 5547.405 110820.210

100366.783 831934.563 386083.791 �50183.392 165597.136

�18782.297 �38432.094 �18654.086 9391.148 �7395.960

�1633.665 �306291.022 �130154.569 816.833 �56015.454

69. ,



1.

3.

5. �121

232

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01�

A15 � � 54611092316384

54621092216384

54611092316384�

A10 � �171341512

170342512

171341512�,A2 � �1

12

022

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A2 � ��2.95�2.8

5.2�1.5

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�1.212.541.110.3

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A2 � �10

01�, A3 � �1

001�, and A4 � �1

001�

A2 � �19

04�, A3 � � 1

2108�, and A4 � � 1

450

16�

A2 � � a1 � a

1 � a�a � � � a

1 � a1 � a

�a � � �10

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AB � �56

�24� � �7

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� 7. 9.

11. 13.

15. 17.

19.

21. ⇒

23. ⇒

25.

27. 29.

31. Inverse does not exist.

33.

35.

37.

39. x � 36, y � �14 41. x � 2, y � 1

43. x � 88, y � �36 45. x � , y � , z �

47. 49. x � � , y � , z �

51. � 0.00540.0104

�0.0193

0.0509�0.0186

0.0116

�0.00660.00950.0344�

169

103

149x � 20

3 , y � 24, z � 563

659

263

149

� 42�9�7

�1843

�511��

213� � � 5

�1

11

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0

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0��1525

25

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1525

35

�15� �

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112

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18

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53.

55.

57. x � 4.5666, y � �6.4436, z � �24.0747

59. x � �1.1874, y � 2.4568, z � 8.2560

61.

63.

65.

Exercise 2.7: Application 1 (p. 136)

1. A’s wages � C’s wages � $30,000; B’s wages � $22,500

3. A’s wages � $12,000; B’s wages � $22,000; C’s wages �$30,000

5.

7. Farmer’s wages � $20,000;

Builder’s wages � $18,000;

Tailor’s wages � $12,000;

Rancher’s wages � $25,000

9.

Exercise 2.7: Application 2

1. a. THIS IS KILLING MY GPAb. EVERYONE LOVES MICKEY

3. WHATS YOUR EMAIL ADDRESS

5. a. 21 11 47 27 57 33 49 28 85 50 35 22 80 47 85 55b. 61 33 101 61 84 50 64 36 49 31 70 45c. 39 22 97 58 59 37 42 27 48 25 31 20 49 28

X � � 16075.384�

X � �203.282166.977137.847�

��158

2�1�

�015

12

� 110�

AA�1 � �ac

bd�� d

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516

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161732

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38

1116 �7

87

16

74 � 3

16 �9132

4716 �23

32

�32

18

4916 �21

85

16

14 � 1

16 � 932

516

332

�0.0249

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�0.0175

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�0.00570.0292

�0.04210.0657

0.0059�0.0305

0.00050.0619

�Exercise 2.7: Application 3

1.

Indirect Costs forServices from

Total Direct DepartmentsCosts Costs, Dollars

Department Dollars Dollars S1 S2

S1 3109.10 2000 345.46 763.63

S2 2290.90 1000 1036.37 254.54

P1 3354.55 2500 345.46 509.09

P2 2790.91 1500 1036.37 254.54

P3 3854.55 3000 354.46 509.09

Totals 15,400.01 10,000 3109.12 2290.89

Total of the service charges allocated to P1, P2, and P3:$3000.01

Sum of the direct costs of the service departments, S1, and S2:$3000

Exercise 2.7: Application 4 (p. 149)

1. 3.

5.

7. a. y � �

b. 17,743 units will be supplied.

9. y � 1.4978x � 36.1345

11. a. Not symmetricb. Symmetricc. Not symmetricYes, for two matrices to be equal, they must have the samedimensions.

275

5435 x

[8 6 3]

� 111

012

14��4

12

310�

CHAPTER 2 Review


1. T 2. F 3. F 4. T 5. F 6. F 7. F


1. 3 � 2 2. one; infinitely many 3. rows; columns

4. inverse 5. 3 � 3 6. 5 � 5

Review Exercises (p .181)

1. x � 2, y � �1

3. x � 2, y � �1

� �


5. No solution, the system is inconsistent.

7. x � �1, y � 2, z � �3

9. x � � , y � � , and z, where z is any real number.

11.

13.

15. x � , y �

17. x � �103, y � 32, z � 9

19. x � 29, y � �10, z � �1


23. x � 9, y � � , z � �

25. x � 29, y � 8, z � �24

27. x � , y � � , and z, where z is any real number.Answers will vary. Three possible solutions are x � , y �� , when z � 0; x � , y � � , when x � 1; and x � 1, y ��2, when z � �1.

29. x � � � 1, y � � 2, and z, where z is any real number.Answers will vary.


33. One solution, x � 11, y � �1, and z � �1.

35. Infinitely many solutions, x1 � �2x4 � 1, x2 � �2x4 � 1,x3 � 3, and x4, where x4 is any real number.

37. The dimensions are 3 � 2.





47. The dimensions are 2 � 3.�10

24

�12�

�122116

�160

�12

8�18

0�� 8

918

�132

�17

8� 10

4�� 4

12�2

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5

0�8�4�

� 612

�6

02412�

�434

�490�

45 z3

5 z

47

137

97

107

97

57 z3

7 z � 107

373

563

269

149

�x � 4y � 6z � �1

�3x � 2y � 8x � 4y � �1

698

98 z

394

74 z

49. . The dimensions are 3 � 2.

51.

53. Inverse does not exist.

55.

57. AB � BA when x � w and y � �z.

59. Each box should contain 20 caramels and 30 creams. Toobtain a profit, increase the number of caramels in each box,(decreasing the number of creams.)

61.

Small Box Medium Box Large Box

0 66.67 33.33

5 60 35

20 40 40

35 20 45

50 0 50

63. a. To attain $2500 per year in income.


$35,000 $5,000 $ 0

$35,500 $4,000 $ 500

$36,000 $3,000 $1,000

$36,500 $2,000 $1,500

$37,000 $1,000 $2,000

$37,500 $ 0 $2,500

b. To attain $3000 per year in income.


$10,000 $30,000 $ 0

$13,000 $24,000 $ 3,000

$16,000 $18,000 $ 6,000

$19,000 $12,000 $ 9,000

$22,000 $ 6,000 $12,000

$25,000 $ 0 $15,000

�1

163

16

� 316

1323

321332

14

�1414

�

�1323

0

1��3

15

�45

�2� � C

Exercise 3.1 AN-17

c.


$ 0 $25,000 $15,000

$ 2,500 $20,000 $17,500

$ 5,000 $15,000 $20,000

$ 7,500 $10,000 $22,500

$10,000 $ 5,000 $25,000

$12,500 $ 0 $27,500

65. a.

b. y � 4096.7742x � 245,870.9677c. The number of people that emigrate is increasing by 4097

(rounding to the nearest person) people each year.

0

245,000

255,000

265,000

275,000

285,000

295,000

5 10 15

d. An estimated 307,323 people will emigrate in 2005.e. An estimated 266,355 people emigrated in 1995.f. y � 4096.7742x � 245,870.9677

67. 275.564 units of A, 98.86 units of B, and 179.548 units of Cshould be produced.

69.

Total Direct Indirect Costs for Department Costs (in Cost Services

Dollars) In Dollars (in Dollars)

S1 1745.45 800 349.09 596.36

S2 5963.64 4000 174.55 1789.09

P1 2445.45 1500 349.09 596.36

P2 2216.37 500 523.64 1192.73

P3 3338.18 1200 349.09 1789.09

Totals 15,709.09 8000 1745.46 5963.63


1. b 2. b 3. d 4. c

CHAPTER 3 Linear Programming: Geometric Approach


1. 3.

5. 7.

9. 11.

x

6

–2

y

6–6

5x + y = 10

10

(0, 0)

–2

x

4

2

–2

y

2

2x + 3y = 6

(0, 0)–1–2–3–4

x

34

2

y

21 3 4

y = 1

(0, 0)

–1–2

x

34

12

–1–2–3–4

y

21 3 5

x = 4

(0, 0)x

y

–4 –2

–2

2

4

–4

42

(1, 1)

x = 0

13. P1 is part of the graph of the system, P2 and P3 are not.

15. P1 and P3 are part of the graph of the system, P2 is not.

17. b 19. c 21. d 23. c

25. Unbounded. Corner points: (2, 0), (0, 2).

27. Bounded. Corner points: (2, 0), (3, 0), (0, 2).

3–2

x

2

–2

y

2x + 3y = 6

x + y = 2

(0, 2)

(3, 0)

(2, 0)

x

y

–4

–2

2

4

–4

42

(0, 2)

(0, 0)

(2, 0)

x + y = 2

y = 0

x = 0

654321–1–2–3

x

32

x + 5y = 5

–1

y

(0, 0)


29. Bounded. Corner points: (2, 0), (5, 0), (2, 6), (0, 8), (0, 2).

31. Bounded. Corner points: (2, 0), (4, 0), ( , ), (0, 4), (0, 2).

33. Unbounded. Corner points: (1, 0), (0, ), (0, 4).

35. Bounded. Corner points: ( , ), (0, 3), (0, 1).

x

y

3x + y = 3

x + 2y = 2

x + y = 4(0, 3)

(0, 1)

35

45( , )

2 4

35

45

y

x

2

3

5

1

–11–1 3 4 5

y = 4

x + 2y = 1

12

x

y

10

8

6

–2–2 10

(0, 4)

(0, 2)

(2, 0) (4, 0)

127

247( , )

3x + y = 12

2x + 3y = 12

x + y = 2

127

247

8642

x

6

10

4

y

(2, 6)

(2, 0)(0, 2)

(0, 8)

(5, 0)

x + y = 2

2x + y = 10

x + y = 8

37.

Corner points: (0, 0), (120, 0), (60, 90), (0, 120).

39. (a)

(b)

Corner points: (0, 0), ( , 0), (0, 40).

41. (a)

(b)

Corner points: (15000, 0), (25000, 0), (15000, 10000).(c) (15000, 0) represents investing $15,000 in treasury bills.

(25000, 0) represents investing $25,000 in treasury bills.(15000, 10000) represents investing $15,000 in treasurybills and $10,000 in corporate bonds.

x

y

(15000, 10000)

(15000, 0) (25000, 0)

x = 15,000

y = 10,000

x + y = 25,000

20,000

5,000

�x � y � 25,000

x 15,000y � 10,000x 0y 0

803

10 20

x

40

30

20

10

y

(0, 0)

(0, 40)

803( , 0)

4x + 3y = 1203x + 2y = 80

�3x � 2y � 804x � 3y � 120

y 0x 0

2201801006020

x

180

60

20

y

(60, 90)(0, 120)

(0, 0)

(120, 0)

3x + 2y = 360

x + 2y = 240

�4x � 8y � 960

12x � 8y � 1440x 0y 0

Exercise 3.3 AN-19

43. (a)

(b)

Corner points: (0, 16), (3, 1), (5, 0).

45. (a)

(b)

Corner points: (0, ), (20, ), (25, 0).



1. Maximum of 38 at (7, 8). Minimum of 10 at (2, 2).


5. Maximum of 55 at (7, 8). Minimum of 14 at any point onthe line segment between (2, 2) and (8, 1).



11. Corner points: (0, 4), (3, 0), (13, 0).

13. Corner points: (0, 0), (15, 0), (5, 10), (0, 10).

15. Corner points: (3, 0), (10, 0), (10, 8), (0, 8), (0, 4).

103

703

15105

x

20

10

y3x + 3y = 70

5x + 4y = 85 2x + 3y = 50

(25, 0)

103(20, )

703(0, )

�5x � 4y 853x � 3y 702x � 3y 50

x 0y 0

4 521

x

16

12

8

4

y

(3, 1)

(0, 16)

(5, 0)

x + 2y = 55x + y = 16

�x � 2y 55x � y 16

x 0y 0

17. Maximum of 14 at (0, 2). 19. Maximum of 15 at (3, 0).

21. Maximum of 56 at (0, 8). 23. Maximum of 58 at (6, 4).

25. Minimum of 0 at (0, 0). 27. Minimum of 4 at (2, 0).

29. Minimum of 4 at (2, 0). 31. Minimum of at (0, ).

33. Maximum of 10 at any point on the line x � y � 10 between(0, 10) and (10, 0). Minimum of at ( , ).


37. Maximum of 40 at (0, 10). Minimum of at ( , ).



43. Maximun of 58 at (4, 5). Minimum of 12 at (0, 2).

45. Maximum of 240 at (3, 10).

47. Maximum of 216 at (2, 10).

49. Produce 90 low-grade packages and 105 high-grade packagesfor a maximum profit of $69.


1. The farmer should plant x acres of soybeans, where 24 �x � 30, and y � 60 � 2x acres of corn for a maximum profitof $9000.

3. She should invest $12,000 in type A bonds and $8000 in typeB bonds for a maximum return of $2400.

5. Manufacture 500,000 of each vitamin for a maximum profitof $75,000.

7. The store should sell 30 microwaves and 20 stoves for a max-imum revenue of $13,000.

9. Purchase 2025 shares of Duke Energy Corp. and 555 sharesof Eastman Kodak for a maximum annual yield of $3267.

11. A child should have 1 of a serving of Gerber Banana PlumGranola and 0.7 of a serving of Gerber Mixed Fruit CarrotJuice for a minimum cost of $1.44.

13. Run the ad for 20 months at the AOL website and 10 monthsat the Yahoo! website to reach a maximum of 2190 millionpeople.

15. Manufacture 25 rolls of low-grade carpet (and no rolls ofhigh-grade carpet) for a maximum income of $2500.

103

103

703

103

103

203

12

32


CHAPTER 3 Review


1. T 2. F 3. T 4. T 5. T 6. F


1. half plane 2. objective 3. feasible

4. bounded 5. corner point


1. 3.

5. All 3 points P1, P2, and P3 are part of the graph of the system.

7. a

9. Bounded.

Corner points: (4. 0), (0. 4), (0. 6)

11. Bounded.

Corner points: (0, .2), ( ), (0, .6)85, 65

864

x

8

6

4

y

3x + y = 6

x + 2y = 4

(0, 6)

(0, 2)65

85( , )

642

x

6

2

y

3x + 2y = 12

x + y = 4

(0, 6)

(4, 0)

(0, 4)

y

x2

–2–2 4 6

4

6

8

10 5x + y = 10

–2–4

x

4

2

–2

–4

y

4

x + 3y = 0

13. Bounded.

Corner points: (2, .0), (4, .0), (2, .3), (0, .3), (0, .4)

15. Maximum of at .

17. Minimum of 20 at (0, 10).

19. Maximum of 40 at (20, .0), ( , ), and at any point on theline. 2x � y � 40 connecting them.

21. Minimum of 20 at (10, 0).

23. Maximum of 235 at (5, 8). Minimum of 60 at (4, .0), (0, .3)and at all points on the line 3x � 4y � 12, connecting them.

25. Maximum of 155 at (5, .4). Minimum of 0 at (0, 0).

27. Maximum of 42 at (9, 8).

29. Maximum of 24 at (8, 8).

31. Minimum of at .

33. Produce 8 pairs of downhill skis and 24 pairs of cross-country skis for a maximum profit of $1760.

35. Katy should buy 7.5 lb of food A and 11.25 lb of food B for aminimum cost of $18.75.

37. Give the child 1.58 servings of Gerber Banana Oatmeal andPeach and no Gerber Mixed Fruit Juice for a minimum costof $1.25.

39. Manufacture 550,000 high-potency vitamins and 250,000thousand calcium-enriched vitamins for a maximum profitof $67,500.

Mathematical Questions from ProfessionalExams

1. b 2. a 3. c 4. c 5. d 6. c 7. c

8. b 9. b 10. a 11. b 12. e 13. c 14. b

�45, 18

5 �485

403

403

�403 , 40

3 �803

864

x

6

2

y

(2, 3)

(2, 0)

(4, 0)

(0, 4)

(0, 3)

3x + 2y = 12

3x + 2y = 6x + 2y = 8

Exercise 4.1 AN-21

Section 4.1 (p. 205)

1. Standard 3. Nonstandard

5. Nonstandard 7. Nonstandard

9. Standard 11. Cannot be modified

13. Cannot be modified

15. Constraints

17. 5x1 � 2x2 � x3 � s1 � 20Slack variables 6x1 � x2 � 4x3 � s2 � 24

x1 � x2 � 4x3 � s3 � 16

and initial simplex tableau

19.

The initial simplex tableau is

21.

The initial simplex tableau is:

23. P � 3x1 � 4x2 � 2x3 � 03x1 � x2 � 4x3 � s1 � 5

x1 � x2 � s2 � 5x1 � x2 � x3 � s3 � 6

x1 0, x2 0, x3 0, s1 0, s2 0, s3 0The initial tableau is:

BVs1

s2

P

�P001

x 1

13

�2

x 2

12

�3

x 3

11

�1

s1

100

s2

010

RHS

5010

0�

x 1 0, x 2 0, x 3 0, s1 0, s2 0

P � 2x 1 � 3x 2 � x 3

x 1 � x 2 � x 3 � s1 � 503x 1 � 2x 2 � x 3 �s2 � 10

BVs1

s2

s3

P

�P0001

x 1

2.20.8

1�3

x 2

�1.81.2

1�5

s1

1000

s2

0100

s3

0010

RHS

52.50.1

0�

x 1 � 0, x 2 � 0, x 3 � 0s1 � 0, s2 � 0, s3 � 0

P � 3 x 1 � 5x 2 � 02.2x 1 � 1.8x 2 � s1 � 50.8x 1 � 1.2x 2 � s2 � 2.5

x 1 � x 2 � s3 � 0.1

BVs1

s2

s3

P

�P0001

x 1

561

�2

x 2

�211

�1

x 3

144

�3

s1

1000

s2

0100

s3

0010

RHS202416

0�

x 1 � x 2 � x 3 � 6�2x 1 � 3x 2 � 12x 3 � 2x 1 0, x 2 0, x 3 0

25. Maximize P � x1 � 2x2 � 5x3

Subject to the constraintsx1 � 2x2 � 3x3 � 10

�3x1 � x2 � x3 � 12x1 0, x2 0 x3 0

System with slack variables:P � x1 � 2x2 � 5x3 � 10

x1 � 2x2 � 3x3 � s1 � 10�3x1 � x2 � x3 � s2 � 12

x1 0, x2 0, x3 0, s1 0, s2 0

Initial tableau:

27. Maximize P � 2x1 � 3x2 � x3 � 6x4

Subject to the constraints�x1 � x2 � 2x3 � x4 � 10�x1 � x2 � x3 � x4 � 8

x1 � x2 � x3 � x4 � 9x1 0, x2 0, x3 0, x4 0

System with slack variablesP � 2x1 � 3x2 � x3 � 6x4 � 10

� x1 � x2 � 2x3 � x4 � s1 � 10� x1 � x2 � x3 � x4 � s2 � 18

x1 � x2 � x3 � x4 � s3 � 19x1 0, x2 0, x3 0, x4 0s1 0, s2 0, s3 0

29. New tableau:

New system:

Current values: P � 300, x 2 � 150, s2 � 180

x 2 � �12 x 1 � 1

2s1 � 150s2 � �2x 1 � s1 � 180P � �s1 � 300

BVx 2

s2

P

�P001

x 112

20

x 2

100

s112

�11

s2

010

RHS150180300

�

BVs1

s2

s3

P

�P0001

x 1

�1�1

1�2

x 2

111

�3

x 3

2�1

1�1

x 4

111

�6

s1

1000

s2

0100

s3

0010

RHS

10890�

BVs1

s2

P

�P001

x 1

1�3�1

x 2

�2�1�2

x 3

�31

�5

s1

100

s2

010

RHS

1012

0�

BVs1

s2

s3

P

�P0001

x 1

312

�3

x 2

�11

�1�4

x 3

401

�2

s1

1000

s2

0100

s3

0010

RHS5560�

CHAPTER 4 Linear Programming: Simplex Method


31. New tableau:

New system:

Current values:

33. New tableau:

New system: s1 � 3x1 � x3 � 20x4 � �2x1 � s2 � 24s3 � 3x2 � x3 � 28s4 � 2x1 � 2x3 � s2

P � �7x1 � 2x2 � 3x3 � 4s2 � 96

Current values: P � 96, s1 � 20, x4 � 24, s3 � 28,s4 � 0


1. (b); the pivot element is 1 in row 1, column 2

3. (a); the solution is P � , x1 � , x2 � 0 5. (c)

7. (b); the pivot element is 1 in row 3, column 4

9. The maximum is P � � when x1 � , x2 � .

11. The maximum is P � 8 when x1 � , x2 � .

13. The maximum is P � 6 when x1 � 2, x2 � 0.

15. There is no maximum for P; the feasible region is unbounded.

17. The maximum is P � 30 when x1 � 0, x2 � 0, x3 � 10.

19. The maximum is P � 42 when x1 � 1, x2 � 10, x3 � 0, x4 � 0.

21. The maximum is P � 40 when x1 � 20, x2 � 0, x3 � 0.

23. The maximum is P � 50 when x1 � 0, x2 � 15, x3 � 5, x4 � 0.

25. The maximum profit is $1500 when the manufacturer makes400 of Jean I, 0 of Jean II, and 50 of Jean III.

27. The maximum profit is $190 from the sale of 0 of product A,40 of product B, and 75 of product C.

23

23

127

247291

7204

7

327

2567

BVs1

x 4

s3

s4

P

�P00001

x 1

�320

�27

x 2

00

�3�3�2

x 3

1010

�3

x 4

01000

s1

10000

s2

010

�14

s3

00100

s4

00010

RHS202428

096�

P � 272 , s1 � 6, s2 � 55

2 , x 3 � 92

s1 � 2x 1 � s3 � 6

s2 � �54x 1 � 3

2x 2 � 14s3 � 55

2

x 3 � �34x 1 � 1

2x 2 � 14s3 � 9

2

P � �54x 1 � 1

2x 2 � 34s3 � 27

2

BV

s1

s2

x 3

P

�P

0

0

0

1

x 1

�2543454

x 2

0

�3212

�12

x 3

0

0

1

0

s1

1

0

0

0

s2

0

1

0

0

s3

�1

�141434

RHS

655292

272

�29. The maximum revenue is $275,000 when 200,000 gal of

regular, 0 gal of premium, and 25,000 gal of super premiumare mixed.

31. The maximum return is $7830 when she invests $45,000 instocks, $31,500 in corporate bonds, and $13,500 in munici-pal bonds.

33. The maximum profit is $14,400 when 180 acres of crop A,20 acres of crop B, and 0 acres of crop C are planted.

35. The maximum revenue is $2800 for 50 cans of can I, no cansof can II, and 70 cans of can III.

37. The maximum profit is $12,000 from 1200 television cabinetsand no stereo or radio cabinets.

39. The maximum profit is $30,000 when no TV are shippedfrom Chicago, 375 TVs are shipped from New York, and noTVs are shipped from Denver.


1. Standard form 3. Not in standard form

5. Not in standard form

7. Maximize P � 2y1 � 6y2

subject to y1 � 2y2 � 2y1 � 3y2 � 3y1 0 y2 0


subject to y1 � 2y2 � 3y1 � y2 � 1y1 � 1y1 0 y2 0


subject to y1 � 3y2 � 3y1 � 3y2 � 4y1 0 y2 0

13. The minimum is C � 6 when x1 � 0, x2 � 2

15. The minimum is C � 12 when x1 � 0, x2 � 4

17. The minimum is C � when x1 � , x2 � 0, x3 �

19. The minimum is C � 5 when x1 � 1, x2 � 1, x3 � 0, x4 � 0

21. Mr. Jones minimizes his cost at $0.22 when he adds 2 of pillP and 4 of pill Q to his diet.

23. Argus Company has a minimum cost of $290 by producing20 units of A, 30 units of B and 150 units of C.

25. Mrs. Mintz minimizes her cost at $65.20 by purchasing 4 ofLunch #1, 3 of Lunch #2, and 2 of Lunch #3.

135

85

215



1. The maximum is P � 44 when x1 � 4, x2 � 8

3. The maximum is P � 27 when x1 � 9, x2 � 0, x3 � 0

5. The maximum is P � 7 when x1 � 1, x2 � 2

7. x1 � 0, x2 � 0, x3 � , z �

9. M1 : A1 : 100, M2 : A1 : 400, M1 : A2 : 300, M2 : A2 : 0,C � $150,000

11. xI � , xII � , xIII � 0, C � $7.50

13. Ship 55 sets from the first warehouse to the first retailer, and75 sets from the second to the second, for a minimum costof $965.

15. The company should send 10 representatives from New Yorkand 5 from San Francisco to Dallas and any combination ofrepresentatives to Chicago (10 from SF; 0 from NY, or 9 fromSF, 1 from NY or 8 from SF, 2 from NY) for a minimum costof $6300.

254

58

203

203

CHAPTER 4 Review

True–False Items

1. T 2. F 3. T 4. F 5. T 6. F

Fill in the Blanks

1. Slack variables 2. column 3.

4. Van Neuman Duality Principle

Review Exercises

1. In standard form

3. In standard form



9.

11.

13. BVs1

s2

P

�P001

x 1

14

�1

x 2

31

�2

x 3

11

�1

x 4

26

�4

s1

100

s2

010

RHS

2080

0�

BVs1

s2

s3

P

�P0001

x 1

151

�6

x 2

531

�3

s1

1000

s2

0100

s3

0010

RHS200450120

0�

BVs1

s2

s3

P

�P0001

x 1

212

�2

x 2

533

�1

x 3

113

�3

s1

1000

s2

0100

s3

0010

RHS100

80120

0�

15. (a) The pivot element 2 is found in row s2, column x2. Thenew tableau after pivoting is

BV P x1 x2 s1 s2 RHS

(b) The resulting system of equations is

x1 � 15 � 4s1 � s2

x2 � 5 � s1 � s2

P � 125 � s1 � s2

(c) The new tableau is final tableau. The solution is maxi-mum P � 125 when x1 � 15 and x2 � 5.

17. (a) The pivot element 1 is the found in row s2, column x3.The new tableau after pivoting is

BV P x1 x2 x3 s1 s2 RHS

(b) The resulting system of equations iss1 � 14 � x1 � 2x1 � s2

x3 � 4 � x2 � s2

P � 12 � 2x1 � 2x2 � 3s2

(c) The problem requires additional pivoting. The new pivot1 is found in row s1, column x1.

19. (a) The pivot element 0.5 is found in row s1, column x1.The tableau after pivoting is

BV P x1 x2 s1 s2 RHS

(b) The resulting system of equations isx1 � 2 � x2 � s1

s2 � 1 � 0.5x2 � s1

P � 5 � 0.5x2 � 5s1

(c) The new tableau is the final tableau. The maximum is P � 5 when x1 � 2 and x2 � 0.

21. (a) The pivot element 1 is found in row s3, column x1. Thetableau after pivoting is

BV P x1 x2 x3 s1 s2 s3 RHS

(b) The resulting system of equations iss1 � 10 � 4x3 � 6s2 � s3

x2 � 8 � 8x3 � 4s2 � s3

x1 � 3 � 3x3 � 5s2 � s3

P � 14 � 13x3 � 15s2 � 3s3

s1

x 2

x 1

P�

0001

0010

0100

483

13

1000

�6�4�5

�15

1113

1083

14�

x 1

s2

P �001

100

10.50.5

2�2

5

010

215�

S1

x 3

P�0

01

10

�2

212

010

100

113

144

12�

72

12

52

x 1

x 2

P�0

0

1

1

0

0

0

1

0

�4

1

1

�52

12

72

15

5

125�


(c) No solution exists since in the pivot column (s2) all 3entries are negative.


25. The maximum is P � 352 when x1 � 0, x2 � 6/5, x3 � 28/5

27. In standard form

29. Not in standard form, constraints are not written as greaterthan or equal to inequalities.

31. Not in standard form, constraints are not written as greaterthan or equal to inequalities.

33. Maximize P � 8y1 � 2y2 subject to the

constraints

y1 0 y2 0

35. Maximize P � 100y1 � 50y2 subjectto the constraints y1 � 2y2 � 5

y1 0 y2 0

37. Minimum is C � 7 when x1 � 3, x2 � 1


y1 � y2 � 4y1 � 3

2y1 � y2 � 22y1 � y2 � 1

41. Maximum is P � 20 when x1 � 0, x2 � 4


45. Maximum is P � 12,250 when x1 � 0, x2 � 5, x3 � 25

47. The brewer should brew no lite beer, 180 vats of regularbeer, and 30 vats of dark beer to attain a maximize profit of$4500.

49. The manufacturer should ship no cars to dealer D2 and 25cars to dealer 2 from warehouse 1, and ship 40 cars to dealer1 and none to dealer 2 from warehouse 2. The minimumcost is $10,150.

51. The farmer will realize a maximum profit of $5714.29 if 0acres of corn, 0 acres of wheat, and 142.85 acres of soy areplanted.

53. The pension fund should purchase 1071.4 shares of DukeEnergy, 666.67 shares of Eastman Kodak, 294.12 shares ofGeneral Motors, 158.73 shares of H.J. Hienz to attain a maxi-mum value of $3250.


(b) The resulting system of equations iss1 � 10 � 4x3 � 6s2 � s3

x2 � 8 � 8x3 � 4s2 � s3

x1 � 3 � 3x3 � 5s2 � s3

P � 14 � 13x3 � 15s2 � 3s3

(c) No solution exists since in the pivot column (s2) all 3entries are negative.


25. The maximum is P � 352 when x1 � 0, x2 � 6/5, x3 � 28/5

27. In standard form

29. Not in standard form, constraints are not written as greaterthan or equal to inequalities

31. Not in standard form, constrains are not written as greaterthan or equal to inequalities

33. Maximize P � 8y1 � 2y2 subject to the

constraints

y1 � 0 y2 � 0

35. Maximize P � 100y1 � 50y2 subjectto the constraints y1 � 2y2 � 5

y1 � 0 y2 � 0

y1 � y2 � 4y1 � 3

2y1 � y2 � 22y1 � y2 � 1

37. Manimum is C � 7 when x1 � 3, x2 � 1

39. Manimum is C � 350 when x1 � 50, x2 � 50

41. Maximum is P � 20 when x1 � 0, x2 � 4


45. Maximum is P � 12,250 when x1 � 0, x2 � 5, x3 � 25

47. The brewer should brew no lite beer, 180 vats of regularbeer, and 30 vats of dark beer to attain a maximize profit of$4500.

49. The manufacturer should ship no cars to dealer D2 and 25cars to dealer 2 from warehouse 1, and ship 40 cars to dealer1 and more to dealer 2 from warehouse 2. The minimumcost is $10,150.

51. The farmer will realize a maximum profit of $5714.29 if 0acrses of corn, 0 acrses of wheat, and 142.85 acrses of soy areplanted.

53. The pension fund should purchase 1071.4 shares of DukeEnergy, 666.67 shares of Easteran Kodak, 294.12 shares ofgeneral motors 158.73 shares of H.J. Hieng to attain a maxi-mum value of $ 3250.


1. 60% 3. 110%

5. 6% 7. 0.25%

9. 0.25 11. 1.00

13. 0.065 15. 0.734

17. 150 19. 18

21. 105 23. 5%

25. 160% 27. 250

29. 333 31. $10

33. $45 35. $150

37. 10% 39. 33.3%

41. 13.3% 43. $1140

45. $1680 47. $1263.16; 10.52%

49. $2380.95; 9.52% 51. $489.00

53. 3 years14

10003

CHAPTER 5 Finance

55. Take the discounted loan at 9% per annum.

57. Take the simple interest loan at 6.3% per annum.

59. Choose the simple interest loan at 12.3%.

61. $995,000

63. $62.20 interest received; interest rate is 1.24%

65. $4.40 interest received; interest rate is 1.06%


1. $1127.27 3. $578.81

5. $826.74 7. $98.02

9. $466.20

11. (a) $1124.86 interest earned: $124.86(b) $1126.16 interest earned: $126.16(c) $1126.82 interest earned: $126.82(d) $1127.27 interest earned: $127.27

13. (a) $1126.49(b) $1195.62(c) $1268.99

Exercise 5.4 AN-25

15. (a) $4438.55 17. (a) 8.16%(b) $3940.16 (b) 4.07%

19. 26.0% 21. 11.5 years

23. Choose b. 25. $917.43; $841.68

27. 5.35% 29. 6.82%

31. 6 % compounded annually

33. 9% compounded monthly 35. $109,400

37. $656.07 39. $10,420

41. $29,137.83 43. $42,640.10

45. yes 47. $18,508.09

49. 3.67%; $8.246 trillion 51. $10,810.76

53. 9.1% 55. 15.3 years

57. $940.90 59. $858.73

61. 22.8 years 63. 11.2 years


1. $1593.74 3. $5073.00

5. $7867.22 7. $6977.00

9. $113,201.03 11. $147.05 per month

13. $1868.68 per quarter 15. $4119.18 per month

17. $200.46 per month 19. $2088.11 per year

21. $62,822.56 23. $9126.56

25. $524.04 per month

27. $22,192.08 per year

14

39. Projected tuition 2012-2013: $ 6361.34Projected tuition 2013-2014: $ 6685.13Projected tuition 2014-2015: $ 7025.41Projected tuition 2015-2016: $ 7383.00Monthly sinking fund payment: $ 521.29


1. $15,495.62 3. $856.60

5. $85,135.64 7. $229,100; $25,239.51

9. $470.73 per month 11. $1719.43 per month

13. $2008.18 per year 15. $25,906.15

17. The monthly payment for the 9% loan is $1342.71 and themonthly payment for the 8% loan is $1338.30. The monthlypayment for the 9% loan is larger. The total interest paid islarger for the 9% loan, $242,813 compared to $161,192. After10 years the equity from the 8% loan is larger, $89,695.33compared to $67,617.64.

19. $55.82 per month

21. (a) $1207.64(b) $36.24 per week

23. (a) $40,000 down payment(b) $160,000 amount of loan(c) $1287.40 per month(d) $303,464(e) 22 years, 4 months(f) $211,823.20

25. $332.79 per month

27. $474.01 per month; $4,752.48 in interest paid

29. 30 year: monthly payment — $966.37; total interest —$235,893.20 15 year: monthly payment — $1189.89; total interest —$102,180.20

31. about 4 years and 8 months

33. Monthly payments are reduced by $128.07. They willpay $38,420.63 less in interest.

35. 62 months; $50,870.77 in interest saved


1. (a) $2930(b) 14 payments(c) 36 payments(d) $584.62

3. (a) $2162 trout(b) 26 months

5. (a) A0 � 0, An � 1.02An�1 � 500(b) 80 quarters, or 20 years(c) $159,738

Sinking Fund Cumulative Accumulated TotalPayment $ Deposit $ Deposits Interest $

1 22,192.08 22,192.08 0 22,192.08

2 22,192.08 44,384.16 1775.37 46,159.53

3 22,192.08 66,576.24 5,468.13 72,044.37

4 22,192.08 88,768.32 11,231.68 100,000.00

29. $205,367.98

31. $16,555.75 per quarter

33. (a) $180,611.12 35. 34 years(b) $2395.33 semiannually

37. Projected cost (2007): $ 22,997.82Projected cost including tax: $ 25,125.11Monthly sinking fund payment: $ 495.78


7. (a) A0 � 150000, An � An�1 � 899.33(b) $149,850.67

(c)

(d) After 4 years and 10 months (58 payments)(e) With the 360th payment (30 years) the loan is paid.(f) $173,758.80(g) a. A0 � 150000, An � An�1 � 999.33

b. $149,750.67

c.

d. After 3 years and 1 month (37 payments)e. With the 279th payment (23 yrs. 3 mos.) the loan is

paid.f. $128,167.43


1. The lease option is preferable.

3. Machine A is preferable.

5. $1086.46

7. $992.26

9. 4.38%

�1 � 0.0612 �

�1 � 0.0612 �

CHAPTER 5 Review


1. T 2. T 3. F 4. F


1. proceeds 2. present value 3. annuity 4. amortized


1. 15 2. 350 5. 21 % 7. 80

9. 2200 11. Dan paid $19.80 in sales tax

13. $52.50 interest is charged. Dan must repay $552.50.

15. Warren must pay $19,736.84 to settle his debt.

37

17. $125.12

19. (c) The 10% per annum compounded monthly loan willcost Mike less.

21. $71.36 23. 5.95%

25. 5.87% compounded quarterly

27. The Corey’s should save $1619.25 per month.

29. (a) The Ostedt’s monthly payments are $2726.10.(b) They will pay $517,830 in interest(c) After 5 years their equity is $117,508.93

31. The monthly payments are $1,049.00. The equity after 10 years is $21,576.

33. Mr. Graff should pay $119,431.51 for the mine.

35. The investor should pay $108,003.49 for the well.

37. Mr. Jones will have saved $10,078.44.

39. Bill must deposit $37.98 every quarter.

41. The monthly payments are $141.22.

43. The effective rate of interest is 9.38%.

45. John will receive $1156.60 every 6 mos. for 15 years.

47. After 30 months there will be $2087.09 in the fund.

49. The student’s monthly payment is $330.74.

51. Purchasing the trucks is the better choice.

53. $9845.45


1. b 2. c

3. b 4. b

5. d 6. a

7. c

Answers to Odd-Numbered Problems AN-27

CHAPTER 6 Sets: Counting Techniques


1. true 3. false

5. false 7. true

9. true 11. {2, 3}

13. {1, 2, 3, 4, 5} 15. Ø

17. {a, b, d, e, f, q}

19. (a) {0, 1, 2, 3, 5, 7, 8}(b) {5}(c) {5}(d) {0, 1, 2, 3, 4, 6, 7, 8, 9}(e) {4, 6, 9}(f) {0, 1, 5, 7}(g) Ø(h) {5}

21. (a) {b, c, d, e, f, g}(b) {c}(c) {a, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}(d) {a, b, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}

23. (a) (b)

(c) (d)

(e) (f)

(g) (h)

25. A � is the set of members of the board of directors of IBMwho are also customers of IBM.

27. A � D is the set of all customers and/or stockholders of IBM.

29. � D is the set of all members of the board of directors ofIBM who are not customers of IBM.

31. M � S is the set of all male students who smoke.

33. � is the set of sophomores, juniors, and seniors togeth-er with the female freshmen.

35. F � S � M is the set of male freshmen who smoke.

37. The subsets of {a, b, c} are Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c},{a, b, c}.


1. c(A) � 6 3. c(A � B) � 3

5. c[(A � B) � A] � 6 7. c(A � B) � 5

9. c(A � B) � 2 11. c(A) � 10

13. 452 cars 15. c(A) � 24

17. c(A � B) � 34 19. c(A � ) � 15

21. c(A � B � C) � 54 23. c(A � B � C) � 3

25. (a) 536 voters are Catholic or Republican.(b) 317 voters are Catholic or over 54.(c) 134 voters are Democtratic below 34 or over 54.

27. (a) 259 were seniors.(b) 455 were women.(c) 227 were on the dean’s list.(d) 76 seniors were on the deans list.(e) 118 seniors were female.(f) 93 women were on the dean’s list.(g) 912 students were in the college.

29. (a) 40 cars had power steering and air conditioning.(b) 35 cars had automatic transmission and air.(c) 40 cars had neither power steering nor automatic trans-

mission.(d) 205 cars were sold in July.(e) 155 cars were sold with automatic transmission or air

conditioning or both.

B

FM

A

UA B

A ∩ B–

UA B

C

(A ∩ B) ∪ C– –

UA B

A ∩ (A ∪ B)

UA B

A ∪ (A ∩ B)

UA B

C

(A ∪ B) ∩ (A ∪ C)

UA B

C

A ∪ (B ∩ C)

UA B

A = (A ∩ B) ∪ (A ∩ B)–

UA B

B = (A ∩ B) ∪ (A ∩ B)–

31. 8; U

A B

RH +

B –

O+

O –

A–

AB+

AB–

B +A+


33. 46 use only one of the three brands.

35. Ø, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d},{a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}. There are 16subsets of {a, b, c, d}.


1. 8 routes 3. 24 models

5. 864 outfits 7. 67,600 license plates

9. 120 arrangements

11. 264 � 104 � 4,569,760,000 user names

13. 26 � 253 � 10 � 93 � 2,961,562,500 user names

15. 360 words without repeated letters; 1296 words

17. 5040 rankings

19. 410 � 215 � 235 � 34,359,738,368 � 3.436 � 1010 ways

21. (a) 6,760,000 different license plates(b) 3,407,040 license plates without repeated digits(c) 3,276,000 license plates with no repeated letters or digits

23. 16 distinguishable car types are produced.

25. 60 types of homes can be built.

27. 256 different numbers can be formed.

29. 125,000 different lock combinations are possible.

31. 8 different paths through the maze.


1. 60 3. 120

5. 90 7. 9

9. 28 11. 42

13. 40,320 15. 1

17. 56 19. 1

21. The ordered arrangements of length 3 formed from the let-ters a, b, c, d, and e are:

abc, abd, abe, acb, acd, ace, adb, adc, ade, aeb, aec, aed,bac, bad, bae, bca, bcd, bce, bda, bdc, bde, bea, bec, bed,cab, cad, cae, cba, cbd, cbe, cda, cdb, cde, cea, ceb, ced,dab, dac, dae, dba, dbc, dbe, dca, dcb, dce, dea, deb, dec,eab, eac, ead, eba, ebc, ebd, eca, ecb, ecd, eda, edb, edc

P(5, 3) � 60

23. 123, 124, 132, 134, 142, 143, 213, 214, 231, 234, 241, 242, 243,312, 314, 321, 324, 341, 342, 412, 413, 421, 423, 431, 432;P(4, 3) � 24

25. 16 two-letter codes 27. 8 three-digit numbers

29. 24 ways 31. 60 three-letter codes

33. 6720 ways

35. 18,278 companies can be on the NYSE.

37. 132,860 ways

39. (a) 720 arrangements.(b) 120 arrangements if S comes first.(c) 24 arrangements if S must come first and Y last.

41. 19,958,400 ways 43. 3,368,253,000 ways

45. 32,760 ways officers can be chosen.


1. 15 3. 21

5. 5 7. 28

9. abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde; C(5, 3) � 10

11. 123, 124, 134, 234; C(4, 3) � 4

13. 35 ways 15. 2380 ways

17. 1140 ways 19. 56 8-bit strings

21. 33,649 23. 90,720 different arrangements

25. 27,720 ways 27. 336 different committees

29. 27,720 ways

31. � 1.157 � 1076

33. 1950 different collections

35. 1,217,566,350 ways to select.

37. 60 differents ways

39. 10,626 different samples

41. P(50,15) � 2.943 � 1024 different guesses.

43. P(10,4) � 5040 different 4 digit numbers (no repeated digits).


1. x5 � 5x 4y � 10x3y 2 � 10x 2 y 3 � xy 4 � y 5

3. x 3 � 9x 2 y � 27 � y 2 � 27y 3

5. 16x 4 � 32x 3y � 24x 2 y 2 � 8xy 3 � y 4

7. 10 9. 405

11. 256 different subsets 13. 1023 non-empty subsets

15. 512 subsets

17.

19. �126 �

� ��77� � �7

6�� 86� � �9

6� � �66� � �7

6� � �86� � �9

6��10

7 � � �97� � �9

6� � ��87� � �8

6�� 96�

100!22! � 13! � 10! � 5! � 16! � 17! � 17!


21.

True–False Items

1. true 2. true 3. false 4. false

5. true 6. false 7. false

Fill in the Blanks

1. disjoint 2. permutation

3. combination 4. Pascal

5. binomial coefficients 6. binomial theorem

7. 40

Review Exercises

1. �, � 3. none of these

5. none of these 7. �, �

9. �, � 11. �, �

13. �, � 15. �, �

17. (a) {3, 6, 8, 9} (b) {6} (c) B(d) B (e) Ø (f) {1, 2, 3, 5, 6, 7, 8, 9}

19. (a) (b)

(c) (d)

(e) (f)

� n � (n � 1)!

(k � 1)!((n � 1) � (k � 1))!� n � �n � 1

k � 1�

k � �nk� � k �

n!k!(n � k)!

�n � (n � 1)!

(k � 1)!(n � k)!

21.

23. The set of all states whose names begin with A or which endwith a vowel.

25. The set of all states whose names end with a vowel and lieeast of the Mississippi River.

27. The set of all states whose names start with an A or end witha vowel and which lie east of the Mississippi River.

29. c(A � B) � 3

31. (a) c(A � B) � 20(b) A and B are disjoint.

33. c(A � B) � 1; c(A � B) � 5

35. (a) 160 cars were sold in June.(b) 35 cars had only power steering.

37. {1}, {1, 2}, {1, 3}, {1, 2, 3} 39. 1

41. 210 43. 12

45. 20 47. 12

49. 9900 51. 45

53. 1 55. 35

57. 9 59. 10 ways

61. 6 ways 63. 72 different styles

65. 1024 different ways to answer

67. (a) 120 words(b) 20 words if order is not important.

69. (a) 525 different committees(b) 1715 different committees

71. 12,441,600 ways 73. 20,790 different committees

75. 240 ways 77. 24 ways

79. The speakers can be ordered 24 ways.

81. (a) 4845 samples will contain only good plums.(b) 5700 samples will contain 3 good plums and 1 rotten

plum.(c) 7805 samples will contain one or more rotten plums.

83. 360 words can be made.

85. 302, 400 words can be made.

87. x4 � 8x3 � 24x2 � 32x � 16

89. 560

UA B

UA B

A ∪ B–

UA B

(A ∩ B) ∪ B–

UA B

B ∩ A–

UA B

C

(A ∪ B) ∩ C

UA B

C

(A ∩ B) ∩ (C )

UB C

(B ∪ C)–––––––


CHAPTER 7 Probability


1. The outcomes from tossing a coin are H (heads) and T(tails). The sample space is {HH, HT, TH, TT}.

3. The outcomes from tossing each coin are H (heads) and T (tails). The sample space is {HHH, HHT, HTH, HTT,THH, THT, TTH, TTT}.

5. The outcomes from tossing each coin are H (heads) and T (tails), and the outcomes from tossing a die are the numbers1, 2, 3, 4, 5, and 6. The sample space is {HH1, HT1, TH1,TT1, HH2, HT2, TH2, TT2, HH3, HT3, TH3, TT3, HH4,HT4, TH4, TT4, HH5, HT5, TH5, TT5, HH6, HT6, TH6,TT6}.

7. The sample space, S � {RA, RB, RC, GA, GB, GC} where R � Red and G � Green.

9. The sample space, S � {AA, AB, AC, BA, BB, BC, CA, CB,CC}

11. The sample space, S � {AA1, AB1, AC1, BA1, BB1, BC1,CA1, CB1, CC1, AA2, AB2, AC2, BA2, BB2, BC2, CA2, CB2,CC2, AA3, AB3, AC3, BA3, BB3, BC3, CA3, CB3, CC3, AA4,AB4, AC4, BA4, BB4, BC4, CA4, CB4, CC4}

13. The sample space, S � {RA1, RB1, RC1, GA1, GB1, GC1,RA2, RB2, RC2, GA2, GB2, GC2, RA3, RB3, RC3, GA3, GB3,GC3, RA4, RB4, RC4, GA4, GB4, GC4} where R � Red andG � Green.

15. c(S) 16 17. c(S) 216 19. c(S) 1326 21. c(S) 676

23. Valid assignments are: 1, 2, 3, 6

25. Assignment 2 should be used.

27. P(H) � , P(T) �

29. P(1) � , P(2) � , P(3) � , P(4) � , P(5) � ,

P(6) �

31. Define W: A white ball is picked. P(W) �

33. Define G: A green ball is picked. P(G) �

35. Define R: A red ball is picked. P(W � R) �

37. P( ) � 39. P(A) �

41. P(E) � 43. P(E) �

45. Define A: an ace is drawn; H: a heart is drawn. P(A � H) �

47. Define S: a spade is drawn. P(S) �

49. Define F: a picture card is drawn. P(F) �

51. Define E: a card with a number less than 6 is drawn. P(E) �

53. Define A: an ace is drawn. P(A) � 1213

513

313

14

152

16

19

118

1123G � R

823

723

323

19

29

19

29

19

29

14

34

55. Define H: A randomly selected person has health insurance.P(H) � 0.845

57. P(H) � 0.5, P(T) � 0.5, answers will vary but the resultsshould be fairly close to the actual probabilities.

59. P(H) � 0.75, P(T) � 0.25, answers will vary but the resultsshould be fairly close to the actual probabilities.

61. P(Red) � , P(Yellow) � , P(White) � , answers will varybut the results should be fairly close to the actual probabilities.


1. 0.84 3. 0.44 5. 24.0% 7. 60.1%

9. P(A) � 0.75 11. P(A � B) � 0.65

13. P(A � B) � 0.5 15. 17. 0.30 19. 0.2

21. (a) P(A � B) � 0.7 (c) P(B � ) � 0.2(b) P(A � ) � 0.3 (d) P( ) 0.3

23. (a) 0.68 (b) 0.58 (c) 0.32

25. (a) P(1 or 2) � 0.57 (e) P(0 or 1) � 0.29(b) P(1 or more) � 0.95 (f ) P(0) � 0.05(c) P(1, 2 or 3) � 0.83 (g) P(1, 2 or 3) � 0.78(d) P(3 or more) � 0.38 (h) P(2 or more) � 0.71

27. P(E) � 29. P(E) � 31. P(E) �

33. The odds for E: 3 to 2; The odds against E: 2 to 3

35. The odds for F: 3 to 1; The odds against F: 1 to 3

37. 1 to 5, 1 to 17, 2 to 7 39. 23 to 27 41. 11 to 4


1. Probability all 5 are defective � 2.83 � 10�4; probability atleast 2 are defective � 0.103.

3. (a) P(3H) � (b) P(0H) �

5. (a) Probability 3 sevens is 0.005.(b) Probability of at least 2 sums of 7 or 11 is 0.126.

7. Probability of a repeated digit is 0.940.

9. Probability of no repeated letters is 0.006.

11. Probability the lists match is .

13. Probability of least 2 of 6 people are born in the same monthis 0.777.

15. Probability at least 2 senators have the same birthday isalmost 1.

17. Probability L precedes E is .

19. Probability the word begins with L is .15

12

124

132

516

12

512

34

A � BBA

118

815

215

13


21. Probability both the Giants and Dodgers are in the playoffs is .

23. Probability the wild card is from the Central division is .

25. Probability of the given bridge hand is 0.005.

27. Probability no two passengers exit on the same floor is .


1. P(E) � 0.5 3. P(E�F) � 0.429 5. P(E � F) � 0.3

7. P( ) � 0.5 9. P(E�F) � 0.25, P(F�E) � 0.5

11. P(F) � 0.5 13. P(E � F) �

15. (a) P(E) � (b) P(F) � 17. P(C) � 0.69

19. P(C�A) � 0.9 21. P(C�B) � 0.2 23. P(E � F) � 0.1

25. P(F �E) � 0.2 27. P(E� ) � 0.667

29. Probability exactly 2 girls, given 1st child is a girl is .

31. P(4H) � ; Yes, if we know the 2nd throw is a head P(4H/H) � .

33. Probability of drawing a heart and then a red is ; the prob-ability of drawing a red and then a heart is .

35. Probability of drawing 1 white and 1 yellow ball is .

37. (a) Probability of drawing a red ace is .(b) Probability of drawing a red ace given an ace was drawn

is .(c) Probability of drawing a red ace given a red was drawn

is .

39. Probability family has more than 2 children given it has atleast one child is 0.625.

41. P(E) � 0.4 43. P(H) � 0.24 45. P(E � H) � 0.1

47. P(G � H) � 0.08 49. P(E �H) � 0.417

51. P(G�H) � 0.333 53. P(E �F) � 0.107, P(F �E) � 0.818

55. (a) P(M) � (e) P(A�M) �

(b) P(A) � (f ) P(F �A � E) �

(c) P(F � B) � (g) P(M � ) �

(d) P(F �E) � (h) P(F � ) �

57. P(E �F) �

59. Define E: Person is a Republican; F: a person voted for the

Democrat. P(E) � ; P(E �F) � .1120

34

223

117373E51

263

3831009B72

1009

213596

3331009

171724

7241009

113

12

126

15

25204

25204

18

116

12

F

23

12

413

E

105512

513

265 61. P(S) � 0.467 63. P(N �E) �


1. P(E � F) � 0.24 3. P(F) � 0.625

5. No, P(E � F) � � � P(E)P(F).

7. (a) P(E �F) � 0.2 (c) P(E � F) � 0.08(b) P(F �E) � 0.4 (d) P(E � F) � 0.52

9. P(E � F � G) �

11. P(E � F) � 0.5 No, P(E � F) � 0.1 � 0.06 � P(E)P(F).

13. No, P(E � F) � � � P(E)P(F).

15. (a) Yes, P(E � F) � � P(E)P(F).

(b) No, P(E � F) � � � P(E)P(F).

17. P(H) � 19. P(4) � 21. P(4) �

23. F(5 � 6) � 25. P[(4 � 5 � 6) � H] �

27. P(RRR) � , P(RRL) � , P(RLR) � , P(LRR) � ,

P(RLL) � , P(LRL) � , P(LLR) � , P(LLL) �

(a) P(E) � (b) P(F) � (c) P(G) � (d) P(H) �

29. P(E � F) � � P(E)P(F)

31. (a) Probability both children have heart disease is .(b) Probability neither child is diseased is .(c) Probability exactly 1 has disease is .

33. (a) P(3T) � (b) P(2H and 1T) �

35. (a) Probability all recover is 0.656.(b) Probability 2 recover is 0.049.(c) Probability at least 2 recover is 0.996.

37. (a) Probability both are red is .

(c) Probability one is red is .

39. (a) P(A�U) � (b) P(A� ) �

(c) No, P(U � A) � � � P(U)P(A).

(d) No, P(U � ) � � � P(U)P( ).

(e) No, P( � A) � � � P( )P(A).

(f ) No, P( � ) � � � P( )P( ).

41. (a) Probability both vote for the candidate is .

(b) Probability neither vote for the candidate is .

(c) Probability one votes for the candidate is .49

19

49

AU54599408

1728AU

U1033136

1168U

A34459408

1956A

653136

121

1103U8

65

1225

925

964

2764

38

116

916

14

12

12

212

14

212

112

212

212

112

112

212

112

14

13

56

16

12

524

14

14

14

16

4147

19

29

511

CHAPTER 7 Review


1. T 2. F 3. F 4. F

5. T 6. F 7. T 8. T


1. 2. 32 3. 1, 0 4. 0.8

5. in favor of 6. equally likely 7. mutually exclusive

12


23. (a) P( ) � 0.70 (e) P( ) � 1(b) P( ) � 0.55 (f) P( ) � 0.25(c) P(E � F) � 0 (g) P( ) � 1(d) P(E � F) � 0.75 (h) P( ) � 0.25

25. P(E � P) �

27. (a) No, since P(0) � � P(1) � � P(2) � .(b) Outcome 0 has the highest probability.(c) P(F) �

29. P(5 � 7 � 9) � 31. 1 to 5

33. Probability Bears win is .

35. P(E � F) � � P(E)P(F)

37. Define E: A student fails mathematics, and F: A studentfails physics. (a) P(E �F) � 0.333; (b) P(F �E) � 0.237 (c) P(E � F) � 0.560

39. The probability is

41. Define E: A person has blue eyes, F: A person has browneyes, G: A person is left handed.

(a) P(E � G) � 0.025 (b) P(G) � 0.0625 (c) P(E �G) � 0.4

43. (a) P(F �E) �

(b) P(E �F) �(c) Let E � scored over 80% and F � took form A.

Yes, because P(E � F) � 0.08 � P(E)P(F).(d) Let E � scored over 80% and F � took form B.

Yes, because P(E � F) � 0.12 � P(E)P(F).

45. Probability at least one matched is

47. (a) Probability all are underweight is 0.0002(b) Probability 2 are underweight is 0.083(c) Probability at most 1 is underweight is 0.910

49. Probability all are born on different days is 0.017

51. P(E �F) � 0.2

53. (a) Probability misses the 1st and gets the next 3 is 0.1029(b) Probability misses 10 in a row is 0.0282

55. Probability the car is black is

Mathematical Questions from Professional Exams (p. 419)

1. b 2. e 3. b 4. d 5. c

6. d 7. b 8. a 9. c 10. b

14

2527

15

25

5216

316

713

512

964

38

18

12

12

E � FE � FE � FFE � FE

CHAPTER 8 Additional Probability Topics


1. P(E � A) � 0.4 3. P(E � B) � 0.2 5. P(E �C) � 0.7

7. P(E) � 0.31 9. P(A�E) � 11. P(C �E) �7

311231

13. P(B �E) � 15. P(E) � 0.024 17. P(E) � 0.016

19. P(A1 � E) � 0.5; P(A2 � E) � 0.5

21. P(A1 � E) � 0.375; P(A2 � E) � 0.375; P(A3 � E) � 0.25

1231


1. S � {0, 1, 2, 3, 4, 5}

3. The outcomes for each child are boy (B) and girl (G). Thesample space is {BB, BG, GB, GG}.

5. P(penny) � , P(dime) � , P(quarter) �

7. P(1) � , P(2) � , P(3) � , P(4) � , P(5) � , P(6) �

9. (a) Let X � the number of girls in a family of 4 children

X P (X)

0 0.0625

1 0.25

2 0.375

3 0.25

4 0.0625

(b) (i) P(0) � 0.0625 (iii) P(1) � 0.25(ii) P(2) � 0.375 (iv) 1 � P(4) � 0.9375

11. (a) Let X � the number of tails observed on three tosses ofa coin

X P (X)

0

1

2

3

(b) (i) P(3) � (iii) P(2) �

(ii) P(0) � (iv) P(2 � 3) �

13. (a) Probability both are blue is .

(b) Probability 1 is blue is .

(c) Probability at least 1 is blue is .

15. (a) P(A � B) � 0.6 (c) P( ) � 0.4(b) P( ) � 0.7 (d) P( � ) � 0.8

17. (a) P(3) � (b) P(5) � (c) P(6) �

19. (a) P( ) � 0.35 (c) No, because P(E � F) � 0.3 � 0.(b) P(E � F) � 0.75

21. (a) P( ) � (b) P(F) � (c) P( ) � 1324F11

2412E

E

73400

23100

21100

BAAA � B

5591

4591

1091

12

18

38

18

18

38

38

18

18

14

18

18

14

18

25

13

415

Exercise 8.3 AN-33

23. P(A1 � E) � 8/9 � 0.276; P(A2 � E) � 20/29 � 0.690;P(A3 � E) � 0.034

25. P(A2 � E) � 0; P(A3 � E) � 0.065;P(A4 � E) � 0; P(A5 � E) � 2/31 � 0.065

27. P(UI � E) � � 0.333; P(UII � E) � � 0.2;

P(UIII � E) � � 0.467

29. P(M �CB) � 0.953

31. P(D �V) � 0.385; P(R �V) � 0.39; P(I �V) � 0.225

33. P(Rock � positive) � 0.385; P(Clay � positive) � 0.209;P(Sand � positive) � 0.405

35. P(R) � 0.466; P(N �R) � 0.343

37. The probability the nurse forgot is .

39. Probability a student majors in engineering given she isfemale is 0.217.

41. (a) Probability a person is diseased given is positive test 0.858.(b) Probability a patient has the disease given a positive test

is 0.503.(c) Probability a patient has the disease given two positive

tests is 0.961.


1. b(7, 4; .20) � 0.0287 3. b(15, 8; .30) � 0.0138

5. b(15, 10; ) � � 0.0916 7. 0.2969

9. b(3, 2; ) � 11. b(3, 0; ) �

13. b(5, 3; ) � 15. b(10, 6; .3) � 0.0368

17. b(12, 9; .8) � 0.2362

19. P(At least 5 successes) � 0.0580

21. b(8, 1; ) � 23. P(at least 5 tails) �

25. P(2H/at least 1H occurs) � 27. b(5, 2; ) �

29. (a) b(8, 1; .05) � 0.2793 (c) 1 � P(8, 0; .05) � 0.3366(b) b(8, 2; .05) � 0.0515 (d) P(Fewer than 3 defective) �

0.9942

31. b(6, 3; .5) � � 0.3125516

6253888

16

28255

93256

132

12

80243

23

125216

16

29

13

300332768

12

911

715

315

515

33. (a)

(b) P(Exactly 2 successes) �

(c) b �

35. P(At least 2 hits) � 0.9996

37. P(At least 10 correct) �

P(At least 12 correct) � 0.6482

39. b(8, 8; .60) � 0.0007 41. b(10, 4; .23) � 0.1225

43. (a) P(At least 5 correct) �(b) P(At least 5 correct) � 0.3446

45. (a) b(10, 4; .124) � 0.0224(b) b(10, 0; .124) � 0.2661(c) P(At most 5 are over 65) � 0.9995

47. k Actual Value of P(k)

0 0.0625

1 0.25

2 0.375

3 0.25

4 0.0625

49. b(8, 3; .5) � 0.21875


1. E � 1.2 3. E � 50,800 fans

5. Mary should pay 80 cents per game.

7. Dave should pay $1.67 to play.

9. The price exceeds the expected value by $0.75.

11. (a) E � $0.75 (b) No, the game is not fair.(c) To make the game fair, a player should lose $2.00 if 1 tail

is thrown.

13. It is not fair; the expected loss is $0.42.

15. The expected loss is 1.2 cents, so Sarah should not play.

764

3092048

54256�4, 2; 14�

54256

SF

S

S

F

S

F

S

F

S

S

F

S

F

F

FSFSFSFSFSFSFSF

34

14

34

14

34

14

34

14

34

14

34

14

34

14

34

14

34

14

34

14

34

14

34

14

34

14

34

14

34

14


17. He should bet $7 to make game fair.

19. Management should choose the second location.

21. E � � 333 times 23. E � 10 light bulbs

25. E � 1 person to have an unfavorable reaction.

27. E � 2.734 tosses.

29. The airline should schedule aircraft A to maximize expectedprofit.


1. Expected number of customers is 9; optimal number of carsis 9 for an expected daily profit of $168.

3.

Group size 2 3 4 5 6 7

Expected 0.403 0.524 0.565 0.574 0.568 0.556Tests perSaved Component

The optimal group size is 5.

13

20006

5. (a) E(X) � $75,000 � 75000(.05x) � 500x where x is thenumber of divers hired.

(b) Hiring two divers maximizes the net gain.

7. The probability the message is correctly received is 0.9647.


1. P(X � 0) � ; P(X � 1) � ; P(X � 2) �

3. P(X � 0) � ; P(X � 1) � ; P(X � 2) � ; P(X � 3) �

5. P(X � 0) � ; P(X � 1) � ; P(X � 2) � ; P(X � 3) �

7. E(X) � 1.2

9. Actual probabilities: P(X � 1) � , P(X � 2) � ,P(X � 3) � , P(X � 4) � , P(X � 5) � ; P(X � 6) � .

11. P(0.1 � X � 0.3) � 0.2

13. P(X � 2) � 12

16

16

16

16

16

16

130

310

12

16

18

38

38

18

14

12

14

CHAPTER 8 Review Exercises


1. T 2. F 3. F 4. T 5. F 6. T


1. Bayes’ formula

2. independent; the same

3. expected value of the experiment

4. a real number

5. expected value

Review Exercises

1. P(E � A) � 0.82 3. P(E � B) � 0.10

5. P(A�E) � 0.9866 7. P(B�E) � 0.0134

9. P(E � A) � 0.5 11. P(E �B) � 0.4 13. P(E �C) � 0.3

15. P(E) � 0.43 17. P(A�E) � 0.4651

19. P(B�E) � 0.4651 21. P(C �E) � 0.0698

23. (a) P(E �G) � (e) P(F �G) �

(b) P(G �E) � (f) P(G �F) �

(c) P(H �E) � (g) P(H �F) �

(d) P(K �E) � (h) P(K �F) � 1342

1169

1742

2264

27

1223

313

913

25. P(C) � 0.163

27. (a) b(5, 0; .20) � 0.3277 (b) b(5, 3; .20) � 0.0512

29. (a) b(12, 12; ) � � 0

(b) (c) 793 to 1255

31. P(At least 3 11’s) � .0016

33. E � 1.5 35. E � $29.52

37. (a) 18 cents would be a fair price for a ticket.(b) Alice paid 56 cents extra for the eight tickets.

39. E � 9.75

41. (a) The expected profit is $713,500

43. E � � 83.33

45. The probability the code is correctly received is 0.9349.

47. (a) Probability pooled test is positive is 1 � (1 � p)30.The expected number of tests in a pooled sample isE � 31 � 30 (1 � p)30.

49. (a) X � {0, 1, 2, 3, 4, 5}(b) P(X � 0) � 0.5838; P(X � 1) � 0.3394;

P(X � 2) � 0.0702; P(X � 3) � 0.0064;P(X � 4) � 0.0003; P(X � 5) � 0

(c) E(X) � 0.5

5006

7932048

1212

12


CHAPTER 9 Statistics


1. discrete 3. continuous 5. continuous 7. discrete

9. discrete 11. continuous

13. A poll should be taken either door-to-door or by means ofthe telephone of a cross section of people from differentparts of the country.

15. 17. Answers will vary. All answers should include a methodto choose a sample in which each member of the populationhas an equal chance of being selected.

19. Answers will vary. 21. Answers will vary.


1. (a)

(b) Northeast has the highest median income.(c) South has the lowest median income.

3. (a)

0

5,000

10,000

15,000

20,000

25,000

30,000

North-east

Mid-west

South West

Fam

ilies

(in

th

ousa

nds

)

0

10,000

20,000

30,000

40,000

50,000

60,000

Med

ian

inco

me

North-east

Mid-west

South West

(b)

(d) The South has the most families.(e) The Northeast has the fewest families.

5. (a)

(b) Married couple families have the highest median income.(c) Female householder — no spouse families have the low-

est median income.

7. (a)

0

100,000

200,000

300,000

400,000

500,000

600,000

700,000

800,000

Hea

rt d

isea

se

Can

cer

Stro

ke

Res

pira

tory

dis

ease

s

Acc

iden

ts

Dia

bete

s

Alz

hei

mer

's

Kid

ney

failu

re

Sept

icem

ia

$0

$15,000

$30,000

$45,000

$60,000

Med

ian

inco

me

Married Femalehouseholder

Malehouseholder

West20.6%

Northeast18.5%

Midwest22.9%South

38.0%

(b)

Heart disease37%

Cancer29%

Stroke9%

Respiratory diseases10%

Accidents5%

Diabetes4%

Alzheimer's3%

Kidney failure2% Septicemia

2%

(d) Heart disease was the leading cause of death in 2000.

9. (a) Sky West had the highest percentage of on-time flights.(b) Atlantic Coast had the lowest percentage of on-time

flights.(c) 84.3% of United Airline’s flights were on time.

11. (a) Housing, fuel and utilities are the largest component ofthe CPI.

(b) Miscellaneous goods and services from the smallestcomponent of the CPI.


1. (a)

25 30 35 40 45 50 550

1

2

3

4

5

6

7

8

9

Freq

uen

cy f

Score

(b)


(c) Class Freq Class Freq Class Freq Class Freq

24 – 25.9 1 32 – 33.9 3 40 – 41.9 6 48 – 49.9 4

26 – 27.9 1 34 – 35.9 3 42 – 43.9 4 50 – 51.9 2

28 – 29.9 2 36 – 37.9 6 44 – 45.9 3 52 – 53.9 5

30 – 31.9 5 38 – 39.9 2 46 – 47.9 3 54 – 55.9 3

Score Freq Score Freq Score Freq Score Freq Score Freq Score Freq

25 1 31 2 36 2 41 5 46 2 51 1

26 1 32 1 37 4 42 3 47 1 52 3

28 1 33 2 38 1 43 1 48 3 53 2

29 1 34 2 39 1 44 2 49 1 54 2

30 3 35 1 40 1 45 1 50 1 55 1

Exercise 9.3 AN-37

(d) (e)

0

2

4

6

Freq

uen

cy

Score

24–

25.9

26–

27.9

28–

29.9

30–

31.9

32–

33.9

34–

35.9

36–

37.9

38–

39.9

40–

41.9

42–

43.9

44–

45.9

46–

47.9

48–

49.9

50–

51.9

52–

53.9

54–

55.9

0

2

4

6

Freq

uen

cy

Score

24–

25.9

26–

27.9

28–

29.9

30–

31.9

32–

33.9

34–

35.9

36–

37.9

38–

39.9

40–

41.9

42–

43.9

44–

45.9

46–

47.9

48–

49.9

50–

51.9

52–

53.9

54–

55.9

(f)

(g)

0

5

10

15

20

25

30

35

40

45

50

55

Cu

mu

lati

ve fr

equ

ency

24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

Score

Cumulative Cumulative Cumulative CumulativeClass Freq Class Freq Class Freq Class Freq

24 – 25.9 1 32 – 33.9 12 40 – 41.9 29 48 – 49.9 43

26 – 27.9 2 34 – 35.9 15 42 – 43.9 33 50 – 51.9 45

28 – 29.9 4 36 – 37.9 21 44 – 45.9 36 52 – 53.9 50

30 – 31.9 9 38 – 39.9 23 46 – 47.9 39 54 – 55.9 53

3. (a) Class Freq Class Freq Class Freq Class Freq

50 – 54.9 1 70 – 74.9 8 90 – 94.9 12 110 – 115.9 0

55 – 59.9 6 75 – 79.9 11 95 – 99.9 2 115 – 119.9 2

60 – 64.9 3 80 – 84.9 2 100 – 104.9 2

65 – 69.9 6 85 – 89.9 12 105 – 109.9 4

(d)


0

5

10

50–

54.9

55–

59.9

60–

64.9

65–

69.9

70–

74.9

75–

79.9

80–

84.9

85–

89.9

90–

94.9

95–

99.9

100–

104.

9

105–

109.

9

110–

114.

9

115–

119.

9

Weight

Freq

uen

cy

(b) (c)

0

5

10

50–

54.9

55–

59.9

60–

64.9

65–

69.9

70–

74.9

75–

79.9

80–

84.9

85–

89.9

90–

94.9

95–

99.9

100–

104.

9

105–

109.

9

110–

114.

9

115–

119.

9

Weight

Freq

uen

cy

(e)

0

15

30

45

60

75

505050 55 60 65 70 75 80 85 90 95 100

105

110

115

120

Weight

Cu

mu

lati

ve fr

equ

ency

Cumulative Cumulative Cumulative CumulativeClass Freq Class Freq Class Freq Class Freq

50 – 54.9 1 70 – 74.9 24 90 – 94.9 61 110 – 115.9 69

55 – 59.9 7 75 – 79.9 35 95 – 99.9 63 115 – 119.9 71

60 – 64.9 10 80 – 84.9 37 100 – 104.9 65

65 – 69.9 16 85 – 89.9 49 105 – 109.9 69

5. (a) There are 13 class intervals.(b) The lower class limit of the 1st class interval is 20 years, the

upper class limit is 24 years.

(c) The class width is 5 years.(d) There are 1,300,000 drivers between the ages of 70 and 84.(e) The interval 30 – 34 years has the most drivers.(f) The interval 80 – 84 years has the fewest drivers.( g )

20–

24

25–

2930

–34

35–

39

40–

44

45–

4950

–54

55–

5960

–64

65–

69

70–

74

75–

7980

–84

400,000

200,000

0

600,000

800,000

1,000,000

1,200,000

1,400,000

Age

Nu

mbe

r of

lice

nse

d dr

iver

s

Exercise 9.3 AN-39

7. Class Freq Class Freq Class Freq Class Freq

20 – 29 1,860,000 40 – 49 1,870,000 60 – 69 1,230,000 80 – 89 220,000

30 – 39 2,400,000 50 – 59 1,420,000 70 – 79 1,030,000

(a) There are 7 class intervals (b) The lower class limit for the last interval is 80 years, the upper class limit is 89 years.

0

250,000

500,000

750,000

1,000,000

1,250,000

1,500,000

1,750,000

2,000,000

2,250,000

2,500,000

Nu

mbe

r of

lice

nse

d dr

iver

s

20–

29

30–

39

40–

49

50–

59

60–

69

70–

79

80–

89

Age

(c) (d)

0

250,000

500,000

750,000

1,000,000

1,250,000

1,500,000

1,750,000

2,000,000

2,250,000

2,500,000

Nu

mbe

r of

lice

nse

d dr

iver

s

20–

29

30–

39

40–

49

50–

59

60–

69

70–

79

80–

89

Age

(e) Cumulative Cumulative Cumulative CumulativeClass Freq Class Freq Class Freq Class Freq

20 – 29 1,860,000 40 – 49 6,130,000 60 – 69 8,780,000 80 – 89 10,030,000

30– 39 4,260,000 50 – 59 7,500,000 70 – 79 9,810,000

(f)

0

1,050,000

2,100,000

3,150,000

4,200,000

5,250,000

6,300,000

7,350,000

8,400,000

9,450,000

10,500,000

Cu

mu

lati

ve n

um

ber

of li

cen

sed

driv

ers

20 30 40 50 60 70 80 90

Age

(c) The class width is 5 years.

0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

400,000

450,000

Nu

mbe

r of

lice

nse

d dr

iver

s

20–

24

25–

29

30–3

4

35–

39

40–

44

45–

49

50–

54

55–

59

60–

64

65–

69

70–

74

75–

79

80–

84

Age

(d)

9. (a) There are 13 class intervals.(b) The lower class limit of the 1st class interval is 20 years,

the upper class limit is 24 years.

Class Freq Class Freq Class Freq Class Freq

11.0– 11.9 0 13.0– 13.9 5 15.0 – 15.4 2 17.0 – 17.4 1

12.0– 12.9 3 14.0– 14.9 6 16.0 – 16.9 3 18.0 – 18.9 0


0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

400,000

450,000

Nu

mbe

r of

lice

nse

d dr

iver

s

20–

24

25–

29

30–

34

35–

39

40–

44

45–

49

50–

54

55–

59

60–

64

65–

69

70–

74

75–

79

80–

84

Age

(e)

(f) Class interval 35 – 39 has the most licensed drivers.(g) Class interval 80 – 84 has the fewest licensed drivers.

11. (a) There are 15 class intervals.(b) The lower class limit for the first interval is 0, the upper

limit is $999.(c) The class width is $1000.

0

15

30

45

60

75

90

105

120

135

150

Nu

mbe

r of

4-y

ear

colle

ges

0–99

9

1000

–19

99

2000

–29

99

3000

–39

99

4000

–49

99

5000

–59

99

6000

–69

99

7000

–79

99

8000

–89

99

9000

–99

99

10,0

00–

10,9

99

11,0

00–

11,9

99

12,0

00–

12,9

99

13,0

00–

13,9

99

14,0

00–

14,9

99

Tuition (in dollars)

0

15

30

45

60

75

90

105

120

135

150N

um

ber

of 4

-yea

r co

llege

s

0–99

9

1000

–19

99

2000

–29

99

3000

–39

99

4000

–49

99

5000

–59

99

6000

–69

99

7000

–79

99

8000

–89

99

9000

–99

99

10,0

00–

10,9

99

11,0

00–

11,9

99

12,0

00–

12,9

99

13,0

00–

13,9

99

14,0

00–

14,9

99

Tuition (in dollars)

(d)

(e)

(f) Tuition between $8000 and $8999 occurs most frequently.

13. (a)

Exercise 9.4 AN-41

0

1

2

3

4

5

6

7

12.0

–12

.9

11.0

–11

.9

13.0

–13

.9

14.0

–14

.9

15.0

–15

.9

16.0

–16

.9

17.0

–17

.9

18.0

–18

.9

Birth rates per 1000 people

Freq

uen

cy

0

1

2

3

4

5

6

7

12.0

–12

.9

11.0

–11

.9

13.0

–13

.9

14.0

–14

.9

15.0

–15

.9

16.0

–16

.9

17.0

–17

.9

18.0

–18

.9

Birth rates per 1000 people

Freq

uen

cy

(b) (c)

Class Freq Class Freq Class Freq Class Freq

0 – 1.9 6 4 – 5.9 4 8 – 9.9 0 12 – 13.9 1

2 – 3.9 5 6 – 7.9 1 10 – 11.9 3

15. (a)

0

1

2

3

4

5

6

7

8

9

Freq

uen

cy

0–1.

9

2.0–

3.9

4.0–

5.9

6.0–

7.9

8.0–

9.9

10.0

–11

.9

12.0

–13

.9

Death rates (per 1000) from HIV related illness

(b)

0

1

2

3

4

5

6

7

8

9

Freq

uen

cy

0–1.

9

2.0–

3.9

4.0–

5.9

6.0–

7.9

8.0–

9.9

10.0

–11

.9

12.0

–13

.9

Death rates (per 1000) from HIV related illness

(c)


1. (a) mean: 31.25(b) median: 30.5(c) no mode

3. (a) mean: 70.4(b) median: 70(c) mode: 55

5. (a) mean: 78.8(b) median: 82(c) mode: 82

7. (a) mean: 73.33(b) median: 77.5(c) mode: 80

9. (a) mean: 31.29 years of age(b) median: 32 years of age(c) mode: 32 years of age

11. The mean cost per share is $109.40.

13. (a) The mean age of a new mother in 2000 was approxi-mately 27.668 years.

(b) The median age of a new mother in 2000 was approxi-mately 27.45 years.

25. (a)


15. (a) The mean sales are approximately $119,966.(b) The median sales are approximately $120,000.

17. (a) The mean age of a licensed driver was approximately44.709 years.

(b) The median age of a licenced driver was approximately42.5 years.

19. The mean tuition in 1992 – 93 was approximately $8053.77.

21. (a) mean $41,300; median $36,000(b) The median describes the 4 clustered salaries well.


1. s � 7.058 3. s � 6 5. s � 13.946

7. mean: � 31.878; standard deviation: s � 7.921.

9. mean 885.333 hours, standard deviation 69.681 hours.

11. (a) Range: 17 years(b) s � 4.5548 years(c) � � 4.4807 years

13. (a) Population; we have all of the mothers represented.(b) The standard deviation is 6.367 years.

15. (a) Population data; all recorded earthquakes are included.(b) The mean magnitude of the earthquakes is 3.278.(c) The standard deviation of the magnitudes of the earth-

quakes recorded in 1998 is 1.382.

17. (a) s � 15.693285 years.(b) � � 15.693282 years.(c) Answers will vary.

19. (a) Population; all colleges of the kind are represented.(b) The standard deviation of the tuition is $3175.58.

21. (a) We expect at least 75% of the outcomes to be between 19and 31.

(b) We expect at least 64% of the outcomes to be between20 and 30.

(c) We expect at least 88.88% of the outcomes to be between16 and 34.

(d) We expect at most 25% of the outcomes to be less than19 or more than 31.

(e) We expect at most 11.11% of the outcomes to be lessthan 16 or greater than 34.

23. We expect at least 889 boxes to have between 0 and 12 defec-tive watches.

25. (a) Population(b) The mean number of births was 3,939,476.83.(c) The standard deviation of births was 58,187.82.(d) Exact; the data are not grouped.


1. � � 8, � � l

x

3. � � 18, � � l

5. (a) z � �0.66(b) z � �0.44(c) z � �0.01(d) z � 1.71(e) z � 2.57(f) z � 3

7. (a) A � 0.3133(b) A � 0.3642(c) A � 0.4989(d) A � 0.3888(e) A � 0.4893(f) A � 0.2734

9. A – 5.48%; B – 21.95%; C – 34.37%; D – 30.13%; F – 8.08%

11. A � 0.3085 13. A � 0.8181

15. (a) 1365 women are between 62 and 66 inches(b) 1909 women are between 60 and 68 inches(c) 1995 women are between 58 and 70 inches.(d) 3 women are taller than 70 inches.(e) 12 women are shorter than 59 inches.

17. (a) Approximately 1 student should weigh at least 142pounds.(b) We would expect 70% of the students to weigh between

124.61 and 135.39 pounds.

19. 57.05% of the clothing can be expected to last between 28and 42 months.

21. (a) Attendance lower than 10,525 will be in the lowest 70%of the figures.

(b) Approximately 77.46% of the attendance figures arebetween 8500 and 11,000 persons.

(c) Approximately 13.36% of the attendance figures differfrom the mean by at least 1500 persons.

23. Kathleen had the highest relative standing.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of heads

Pro

babi

lity


CHAPTER 9 Review

True–False Items (p. 527)1. False 2. Truee 3. Falsee 4. Truee 5. False

Fill in the Blanks (p. 527)1. (a) mean (b) median (c) mode

2. the standard deviation

3. bell 4. Z-score

5. � � k; � � k

Review Exercises (p. 527)1. circumference, continuous

3. number of people, discrete

5. number of defective parts, discrete

7. Answers will vary. All answers should include a method tochoose a sample of 100 students from the population inwhich each student has an equal chance of being chosen.

9. Answers will vary. All answers should give examples of possi-ble bias.

–3 3

1

–1

(b) Answers vary.(c) mean 4.5, standard deviation 1.775

27. The approximate probability that there are between 285 and315 successes is 0.7372.

29. The approximate probability of obtaining 300 or more suc-cesses is 0.5.

31. The approximate probability of obtaining 325 or more suc-cesses is 0.0307.

33. (a) The approximate probability of having at least 80 but nomore than 90 hits is 0.2286.

(b) The approximate probability of having 85 or more hitsis 0.0918.

35. The approximate probability of selecting at least 10 unsealedpackages is 0.0116.

11. (a)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Ride alone Car pool Ride bus Other

64%

5%

30%

1%

How Chicagoans get to work

Ride alone64%

Ride bus30%

Other1%

Car pool5%

How Chicagoans get to work(b)

37.

The graph assumes its maximum at x � 0.

39. P[X � 10] � 0.0311;

(b)

(c)


(b)

15. (a) American Indian made up the smallest percentage of 4-year college enrollment.

(b) Asian-Americans were overrepresented in 4-year collegesin 1997.

(c) Approximately 533,820 Hispanic students were enrolledin 4-year colleges in 1997.

0

50

100

150

200

250

300

350

400

450

Freq

uen

cy

Do not like Like Like very much

Like36%

Do not like42%

Likevery much

22%

Score Frequency Score Frequency Score Frequency Score Frequency

21 2 62 1 74 1 87 2

33 1 63 2 75 1 89 1

41 2 66 2 77 1 90 2

42 1 68 1 78 2 91 1

44 1 69 1 80 4 92 1

48 1 70 2 82 1 95 1

52 2 71 1 83 1 99 1

55 1 72 2 85 2 100 2

60 1 73 2

0

1

2

3

4

20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Freq

uen

cy f

Test scores

0

5

10

Freq

uen

cy f

20–

29.9

30–

39.9

40–

49.9

50–

59.9

60–

69.9

70–

79.9

80–

89.9

90–

99.9

100–

109.

9

Test score

17. (a) High school diploma represents the highest level of edu-cational attainment of most Americans in 2000.

(b) Approximately 45,000,000 Americans have at least abachelor's degree.

(c) Approximately 32,000,000 Americans do not have a highschool diploma.

(d) Approximately 48,000,000 Americans have gone to col-lege but do not have a bachelor’s degree.

13. (a)

The range is 79.

19. (a)


(d)

0

5

10

Freq

uen

cy f

20–

29.9

30–

39.9

40–

49.9

50–

59.9

60–

69.9

70–

79.9

80–

89.9

90–

99.9

100–

109.

9

Test score

(e)

Cumulative Cumulative Cumulative CumulativeScore Frequency Score Frequency Score Frequency Score Frequency

21 2 62 13 74 27 87 41

33 3 63 15 75 28 89 42

41 5 66 17 77 29 90 44

42 6 68 18 78 31 91 45

44 7 69 19 80 35 92 46

48 8 70 21 82 36 95 47

52 10 71 22 83 37 99 48

55 11 72 24 85 39 100 5

60 12 73 26

(f)

0

10

20

30

40

50

Cu

mu

lati

ve fr

equ

ency

10 20 30 40 50 60 70 80 90 100 110

Test score

(b)

(c) (d)


21. (a)

0

1

2

3

4

Freq

uen

cy f

4'12

"

4'16

"

4'22

"

4'30

"

4'36

"

4'39

"

4'40

"

4'46

"

4'50

"

4'52

"

4'56

"

5'01

"

5'02

"

5'06

"

5'08

"

5'12

"

5'18

"

5'20

"

5'31

"

5'37

"

5'40

"

5'43

"

5'48

"

5'50

"

5'55

"

6'01

"

6'02

"

6'10

"

6'12

"

6'30

"

6'37

"

6'40

"

7'15

"

7'05

"

Time

0

5

10

15

Freq

uen

cy

4'00

–4'

29"

4'30

–4'

59"

5'00

–5'

29"

5'30

–5'

59"

6'00

–6'

29"

6'30

–6'

59"

7'00

–7'

29"

Time

Class FrequencyInterval Fi

4�00–4�29 3

4�30 –4�59 10

5�00–5�29 11

5�30–5�59 9

6�00–6�29 4

6�30–6�59 3

7�00–7�29 2

Time Freq. Time Freq. Time Freq. Time Freq. Time Freq

4�12 1 4�46 2 5�08 1 5�43 1 6�12 1

4�15 1 4�50 1 5�12 2 5�48 1 6�30 1

4�22 1 4�52 1 5�18 1 5�50 1 6�32 1

4�30 2 4�56 1 5�20 3 5�55 1 6�40 1

4�36 1 5�01 1 5�31 2 6�01 1 7�05 1

439 1 5�02 1 5�37 1 6�02 1 7�15 1

4�40 1 5�06 2 5�40 2 6�10 1

The range is 3 minutes, 3 seconds.


(e)

0

5

10

15

Freq

uen

cy

4'00

–4'

29"

4'30

–4'

59"

5'00

–5'

29"

5'30

–5'

59"

6'00

–6'

29"

6'30

–6'

59"

7'00

–7'

29"

Time

Class CummulativeInterval Frequency

4�00–4�29 3

4�30 –4�59 13

5�00–5�29 24

5�30–5�59 33

6�00–6�29 37

6�30–6�59 40

7�00–7�29 42

(f)

(g)A

0

9

18

27

36

Cu

mu

lati

ve fr

equ

ency

4'00

"

4'30

"

5'00

"

5'30

"

6'00

"

6'30

"

7'00

"

7'30

"

Time

Age Freq Age Freq Age Freq Age Freq Age Freq

24 5 29 2 32 8 35 2 38 1

27 1 30 3 33 1 36 1 40 1

28 1 31 2 34 1 37 1 41 1

23. (a)

(b)

0

2

4

6

8

Freq

uen

cy f

24 28 32 36 40

Age

(d)

(e)


Class Freq Class Freq Class Freq

20.0 – 24.9 5 30.0 – 34.9 15 40.0 – 44.9 2

25.0 – 29.9 4 35.0 – 39.9 5

0

3

6

9

12

15

18

Freq

uen

cy

20.0

–24

.9

25.0

–29

.9

30.0

–34

.9

35.0

–39

.9

40.0

–44

.9

Age

0

3

6

9

12

15

18

Freq

uen

cy

20.0

–24

.9

25.0

–29

.9

30.0

–34

.9

35.0

–39

.9

40.0

–44

.9

Age

Cumulative CumulativeClass Frequency Class Frequency

20.0 – 24.9 5 35.0 – 39.9 29

25.0 – 29.9 9 40.0–44.9 31

30.0 – 34.9 24

(g)

25. (a)

0

4

8

12

16

20

24

28

32

20 25 30 35 40 45

AgeC

um

ula

tive

freq

uen

cy

0%

10%

20%

30%

40%

50%

0–$1

2,00

0

$12,

000–

$46,

700

$46,

700–

$112

,850

$112

,850

–$1

71,9

50

$171

,950

–$3

07,0

60

over

$30

7,06

0

Income

Mar

gin

al t

ax r

ate

(c)

(f)

There are 5 class intervals.

Exercise 10.1 AN-49

(b)

0%

10%

20%

30%

40%

50%

0–$1

2,00

0

$12,

000–

$46,

700

$46,

700–

$112

,850

$112

,850

–$1

71,9

50

$171

,950

–$3

07,0

60

over

$30

7,06

0

Income

Mar

gin

al t

ax r

ate

27. (a) Mean: 5.7273 (b) Median: 5 (c) Mode: 0, 4, 8 and 10(d) Range: 12 (e) Standard deviation: 8.4853

29. (a) Mean: 16.2 (b) Median: 7 (c) Mode: 7(d) Range: 98 (e) Standard deviation: 29.5515

31. (a) Mean: 7 (b) Median: 7 (c) Mode: 7(d) Range: 11 (e) Standard deviation: 3.6515

37. (a) Answers may vary. We assume they are a sample of Joe’sscores calculating parts (b) and (c).

(b) Joe’s mean score is 75.57.(c) The standard deviation of Joe’s scores is 2.99.

39. (a) The approximate mean age of a male in 2000 was 37.8years.

(b) The approximate median age of a male in 2000 was41.069 years.

(c) The approximate standard deviation of the ages of malesin 2000 was 23.065 years.

41. We expect at least 75%, or 750 jars, to have between 11.9 and12.1 ounces of jam.

43. The probability a bag weighs less than 9.5 or more than 10.5pounds is less than 0.25.

45. z � �0.667

47. z � 1.4

49. z � 1.667

51. A � 0.0855

53. A � 0.7555

55. (a) 68.26% of the scores are between 20 and 30.(b) 2.28% of the scores are above 35.

57. 0.17% of the dogs will die before reaching the age of 10years, 4 months.

59. Bob scored equally well on both exams.

61. There is a probability of 0.9544 that this week’s productionwill lie between 30 and 50..

63. The probability of obtaining more than 160 positive resultsis approximately 0.001.

65. The probability that in a group of 200 test-takers between110 and 125 pass the test is 0.6893.

CHAPTER 10 Markov Chains; Games


1. (a) The entry represents the probability that an object instate 2 will move to state 1.

(b) v(1) � , v(2) �

(c) v(1) � , v(2) �

3. v(2) � 5. v(1) � [0.525 0.15 0.35]

7. a � 0.4, b � 0.1, c � 1 9. v(5) � [0.5040 0.4960]

11. (b) R C

P �

(c) P 2 � , P 3 � �0.742750.128625

0.257250.871375��0.815

0.09250.1850.9075�

RC

�0.900.05

0.100.95�

�157576

419576�

�1348

3548��14

34�

� 518

1318��13

23�

13. (b) 0 0 0 0 0 0 0

1 0 0 0 0 0 00 1 0 0 0 0 0

0 0 0 0 0 0P � 0 0 0 0 0 0

0 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 1

15. 57% of wine drinkers will be drinking wine X after 2 months.

17. v(10) � [0.3200 0.2892 0.1631 0.2277]

13

13

13

13

13

13

12

12

12

12

12

12

� �


19. (a)

P �

(b) v(0) � [0 1 0 0 0 0](c) v(0)P(10) � [0 0.018752 0 0.012498 0 0.96875](d) Room 6


1. No, when you multiply row 2 by column 1 you will alwaysget a 0. So p21 � 0 for every power of P.


5. Yes, because P2 � , has all positive entries.


9. Yes, because P2 � , has all positive entries.

11. t � 13. t � 15. t �

A B C17. P �

� [0.30769 0.46154 0.23077] In the long run,the grocers’ stock is 30.8% brand A, 46.2% brand B, and23.1% brand C

19. The probability the grandson of a Laborite votes Socialist is 0.09. In the long run the membership distribution is[0.553 0.383 0.064].

21. (a) P �

(b) The probability is 44%.(c) The probability is 38.2%.(d) The distribution of grandchildren’s educational

attainment will be [0.538 0.383 0.079](e) The long run distribution will be [0.649 0.298 0.053]

23. t � [0.125 0.875] 25. t �

27. t � [0.268 0.209 0.021 0.318]

�4929

13�

�0.80.40.2

0.180.50.6

0.020.10.2 �

� 413

613

313�

ABC

�0.70.10.2

0.150.80.2

0.150.10.6 �

�1543

1243

1643��25

35��37

47�

�13135

1813

3163

16163

16

134813285

181348

5245

245

185

24

�

�7

3618

524

1336

3813

4912

1124

�

�12

0012

00

012

0000

0012

000

12

0012

00

012

0000

0012

011

�Exercise 10.3 (p. 565)

1. Yes, state 1 is an absorbing state. 3. No.

5. Yes, states 1 and 3 are absorbing states. 7. No.

9. No. 11. Yes, states 2 and 3 are absorbing states.

13. 15.

17.

19. (a) A person starting with $1.00 is expected to have $3.000.5 time. A person starting with $2.00 is expected tohave $3.00 one time.

(b) A player starting with $3.00 can expect to play 3 gamesbefore absorption.

21. The probability of accumulating $3.00 if he starts with $1.00is . The probabilities of accumulating $3.00 if he starts with$2.00 is .

23. (a) Colleen can expect to place 1.4 wagers before the gameends.

(b) The probability Colleen is wiped out is 0.84.(c) The probability Colleen wins the amount needed to buy

the car is 0.16.

25. (a) The expected number of wagers to be placed before thegame ends is 1.6.

(b) The probability Colleen is wiped out is 0.64.(c) The probability Colleen wins the amount needed to buy

the car is 0.36.

27. On the average a stock showing no change will remain 6.993days in that state.

29. I3 � ; S � ;

Q �

T � � , TS �


1. Katy’s game matrix is one twofinger fingers

.1 finger2 fingers

��11

1�1�

�0.45

0.45

0.75

0.55

0.55

0.25��

1.053

0

0

2

2

0

0.6

0.6

1��

2019

0

0

2

2

0

3535

1�

�0.0500

0.950.50

00.30 �

�000.75

00.20.25��1

00

010

001�

1019

419

�1012

00

0114

012

00012

0

00012

0

0014

012

��

1014

0

0114

0

001412

001412

��1013

0113

0013�


3. Katy’s game matrix is 1 4 7

5. The game is strictly determined; the value is �2.

7. The game is strictly determined; the value is 3.

9. The game is not strictly determined.

11. The game is strictly determined; the value is 2.


15. The game is strictly determined if 0 a 3.

17. a � b � 0 for the game to be strictly determined.


1. E � $1.42 3. E(P,Q)’s 5. E(P,Q)’s 7. E(P,Q)’s

9. E(P,Q)’s 11. E(P,Q)’s 0


1. The optimal strategy for Player I is P � . The optimal

strategy for Player II is Q � . The expected payoff is

E � � 1.75.74

�1434�

�34 14�

13

179

198

94

147

��25

�8

5�811

�811

�14�3. The optimal strategy for Player I is P � . The optimal

strategy for Player II is Q � . The expected payoff is E � .

5. The optimal strategy for Player I is P � . The optimal

strategy for Player II is Q � . The expected payoff is E � .

7. The optimal strategy for the Democrat is to spend 37.5% ofthe time on domestic issues and 62.5% on foreign issues.The optimal strategy for the Republican is to spend 50% ofthe time on domestic issues and 50% of the time on foreignissues. The expected payoff is E � 1.5, so the Democrat gainsat least 1.5 units by employing the optimal strategy.

9. The optimal strategy for the spy is to try the deserted exit8.5% of the time and try the heavily used exit 91.5% of thetime. The optimal strategy for the spy’s opponent is to waitat the deserted exit 22.5% of the time and wait at the heavilyused exit 77.5% of the time. The expected payoff is E � �0.704, favoring the spy.

5071

78�5

838�

�58 38�

13�1

323�

�16 56�

CHAPTER 10 Review

Ture–False Items (p. 581)

1. F 2. F 3. F 4. T 5. T 6. T 7. T


1. 1 � m 2. nonnegative, one 3. v(k�1), v(0)

4. positive 5. payoff


1. Not regular. 3. Not regular.

5. Yes, because P2 �

7. t � 9. t �

11. t � � [0.608 0.127 0.266]

13. (a) The situation froms a Markov chain because the shifts inmarket shares can be thought of as a sequence of experi-ments each of which results in one of a finite number ofstates.

�4879

1079

2179�

�3814

38��47

37�

�83

24059

1803

10

10724071

1803

10

5245

1825

�.

A B C

P �

(b) After one year, A will have 46.7% of the market, B willhave 28.3% of the market, and C will have 25% of themarket.

(c) After two years, A will have 47.2% of the market, B willhave 26.9% of the market, and C will have 25.9% of themarket.

(d) In the long run, A will have 47.3% of the market, B willhave 26.6% of the market, and C will have 26.0% of themarket.

15. (a) The salespersons movement among the universities canbe thought of as a sequence of experiments each ofwhich ends in one of a finite number of states.

(b) v(1) � �125

121

12�

P � U1

U2

U3�

U1

03434

U2

1014

U3

014

0�

ABC

�0.50.40.5

0.20.40.25

0.30.20.25�


(c) In the long run, she should sell at U1 42.9% of the time,at U2 45.7% of the time, and at U3 11.4% of the time.

17. (a) The Markov chain is not absorbing.

19. (a) The Markov chain is absorbing.(b) The absorbing state is state 2.

(c) P �


23. (a) The Markov chain is absorbing(b) The absorbing states are states 1 and 4.

(c)

25. (a)

P � �10.550000

000.55000

00.4500.5500

000.4500.550

0000.4500

00000.451

��

100.30.25

010.10.2

000.50.35

000.10.2�

�11314

0038

02338�

(b) Given that the man started with $2.00, on the averagethe process will be in state one 1.298 times; in state two2.361 times; in state three 1.412 times and in state four0.635 times.

(c) The expected length of the game is 5.71 bets.

(d) The probability the man loses all his money is 0.714;The probability he wins $5 is 0.286.


29. The game is strictly determined; the value of the game is 9.

31. The game is strictly determined. The value of the game is 12.

33. E � 35. E � 1.75 37. E �

39. The optimal strategy for player I is P � . The optimal

strategy for player II is Q �

41. (a) The investor should invest in A 41.9% of the time andinvest in B 58.1% of the time.

(b) The percentage gain is 8.37%

43. The builder should use of the land for the shopping centerand of the land for the houses.1

3

23

�1212�

[12

12]

43

13

CHAPTER 11 Logic and Logic Circuits


1. Proposition

2. Not a proposition

5. Proposition

7. Proposition

9. A fox is not an animal.

11. I am not buying stocks.

13. Someone wants to buy my house.

15. Everybody has a car.

17. Either John is an economics major, or John is a sociologyminor (or both).

19. John is an economics major and a sociology minor.

21. Either John is not an economics major or John is not a soci-ology minor (or both).

23. Either John is not an economics major or John is a sociologyminor (or both).


1.

p q �q p � �q

T T F T

T F T T

F T F F

F F T T

3.

p q �p �q �p � �q

T T F F F

T F F T F

F T T F F

F F T T T


(c) In the long run, she should sell at U1 42.9% of the time,at U2 45.7% of the time, and at U3 11.4% of the time.


19. (a) The Markov chain is absorbing.(b) The absorbing state is state 2.

(c) P �


23. (a) The Markov chain is absorbing(b) The absorbing states are states 1 and 4.

(c)

25. (a)

P � �10.550000

000.55000

00.4500.5500

000.4500.550

0000.4500

00000.451

��

100.30.25

010.10.2

000.50.35

000.10.2�

�11314

0038

02338�

(b) Given that the man started with $2.00, on the averagethe process will be in state one 1.298 times; in state two2.361 times; in state three 1.412 times and in state four0.635 times.

(c) The expected length of the game is 5.71 bets.

(d) The probability the man loses all his money is 0.714;The probability he wins $5 is 0.286.


29. The game is strictly determined; the value of the game is 9.

31. The game is strictly determined. The value of the game is 12.

33. E � 35. E � 1.75 37. E �

39. The optimal strategy for player I is P � . The optimal

strategy for player II is Q �

41. (a) The investor should invest in A 41.9% of the time andinvest in B 58.1% of the time.

(b) The percentage gain is 8.37%

43. The builder should use of the land for the shopping centerand of the land for the houses.1

3

23

�1212�

[12

12]

43

13

CHAPTER 11 Logic and Logic Circuits


1. Proposition


5. Proposition

7. Proposition

9. A fox is not an animal.

11. I am not buying stocks.

13. Someone wants to buy my house.

15. Everybody has a car.

17. Either John is an economics major, or John is a sociologyminor (or both).

19. John is an economics major and a sociology minor.

21. Either John is not an economics major or John is not a soci-ology minor (or both).

23. Either John is not an economics major or John is a sociologyminor (or both).


1.

p q �q p � �q

T T F T

T F T T

F T F F

F F T T

3.

p q �p �q �p � �q

T T F F F

T F F T F

F T T F F

F F T T T

Exercise 11.2 AN-53

5.

p q �p �p � q �(�p � q)

T T F F T

T F F F T

F T T T F

F F T F T

15.

p q r �q p � �q (p � �q)—� r

T T T F F T

T T F F F F

T F T T T F

T F F T T T

F T T F F T

F T F F F F

F F T T F T

F F F T F F

13.

p q �p �q p � q �p � �q (p � q) � (�p � �q)

T T F F T F T

T F F T F F F

F T T F F F F

F F T T F T T

11.

p q �q p—� q p � �q (p

—� q) � (p � �q)

T T F F F F

T F T T T T

F T F T F F

F F T F F F

17.

p p � p p � p

T T T

F F F

7.

p q �p �q �p � �q �(�p � �q)

T T F F F T

T F F T T F

F T T F T F

F F T T T F

9.

p q �q p � �q (p � �q) � p

T T F T T

T F T T T

F T F F F

F F T T F

19.

p q r p � q (p � q) � r q � r p � (q � r)

T T T T T T T

T T F T F F F

T F T F F F F

T F F F F F F

F T T F F T F

F T F F F F F

F F T F F F F

F F F F F F F

(p � q) (p � q) � r q � r p � (q � r)

T T T T

T T T T

T T T T

T T F T

T T T T

T T T T

F T T T

F F F F


(ii) The negation of “Smith is an ex-convict and Smith isrehabilitated” is “Either Smith is not an ex-convict orsmith is not rehabilitated.”

35. (p � q) � r ≡ r � (p � q) (commutative property)≡ (r � p) � (r � q) (distributive property)≡ (p � s) � (q � s) (cummutative property)

39. Either Mike cannot hit the ball well or he cannot pitch strikes.

41. The baby is not crying and the baby is not talking all the time.


1. Converse: q Q �p; contrapositive: �q Q p; inverse: p Q �q

3. Converse: �p Q �q; contrapositive: p Q q; inverse: q Q p

5. Converse: If the grass is wet then it is raining.Contrapositive: If the grass is not wet then it is not raining.Inverse: If it is not raining then the grass is not wet.

7. Converse: If it is not cloudy then it is not raining.Contrapositive: If it is cloudy then it is raining. Inverse: If itis raining then it is cloudy.

9. Converse: If it is cloudy then it is raining. Contrapositive: Ifit is not cloudy then it is not raining. Inverse: If it is not rain-ing then it is not cloudy.

11. (a) If Jack studies psychology then Mary studies sociology.(b) If Mary studies sociology then Jack studies psychology.(c) If Jack does not study psychology then Mary studies

sociology.

31. The proposition “Smith is an ex-convict” is equivalent tothe proposition “Smith is an ex-convict and Smith is an ex-convict”; the proposition “Smith is an ex-convict” is equiva-lent to the proposition “Smith is an ex-convict or Smith is anex-convict”.

33. (i) The negation of “Smith is an ex-convict or Smith is reha-bilitated” is “Smith is not an ex-convict and Smith is notrehabilitated.”

21.

p q p � q p � q p � (p � q) p � (p � q)

T T T T T T

T F F T T T

F T F T F F

F F F F F F

23.

p q �q �q � q p � (�q � q)

T T F T T

T F T T T

F T F T F

F F T T F

(b) p Q (q � r) ≡ �p � (q � r) (hint)≡ (�p � q) � r (associative property)≡ �(p � �q) � r (DeMorgan’s property)≡ (p � �q) Q r (hint)

13. (a)

p q r �q q � r p Q (q � r) p � �q (p � �q) Q r

T T T F T T F T

T T F F T T F T

T F T T T T T T

T F F T F F T F

F T T F T T F T

F T F F T T F T

F F T T T T F T

F F F T F T F T

29.

(p � q) � [(p � q) �p q �p �q p � q �p � �q (�p � �q) (�p � �q)] � p

T T F F T F T T

T F F T F F F F

F T T F F F F F

F F T T F T T F

27.

p q �p q � �p p � (q � �p)

T T F F F

T F F F F

F T T T F

F F T F F

25.

p �p �(�p)

T F T

F T F

Exercise 11.5 AN-55

29. p Q q

31. �q � �p

33. q Q p


1. Let p and q be the statements, p: It is raining, q: John is goingto school. Assume that p Q �q and q are true statements.

Prove: �p is true.Direct: p Q �q is true.

Also, its contrapositive q Q �p is true and q is true.Then, �p is true by the law of detachment.

Indirect: Assume �p is false.Then p is true; p Q �q is true,Meaning, �q is true by the law of detachment.But q is true, and we have a contradiction.The assumption is false and �p is true.

3. Let p, q, and r be the statements, p: Smith is elected presi-dent; q: Kuntz is elected secretary; r: Brown is elected trea-surer. Assume that p Q q, q Q �r and p are true statements.

Prove: �r is true.Direct: p Q q and q Q �r are true.

So, p Q �r is true by the law of syllogism, and p is true.Thus �r is true by the law of detachment.

Indirect: Assume �r is false.Then r is true; p Q q is true; q Q �r is true.So, p Q �r is true by the law of syllogism.r Q �p, its contrapositive, is true.So, �p is true by the law of detachment.But p is true, and we have a contradiction.The assumption is false and �r is true.

5. Invalid. If the hypotheses p Q q and �p are both true, thenp is false. In the conditional, if p is false, q can be either trueor false. So the conclusion, �q, could be true or false.

7. Valid. Let p: Tami studies, q: Tami fails, and r: Tami playswith dolls often. We are given p Q �q and � r Q p and wemust show q Q r. Using the Law of the contrapositive wehave q Q �p and �p Q r. Then by the Law of Syllogism,q Q r.


1. The output is 1 when a) both p � 1 and q � 1 or b) both p � 0, and q � 0, or c) both p � 0, and r � 1.

27.

p q p � q p � (p � q) p � (p � q) ⇔ p

T T T T T

T F T T T

F T T F T

F F F F T

15.

p q �p p � q �p � (p � q)

T T F T T

T F F F F

F T T F T

F F T F T

17.

p q �p �p � q p � (�p � q)

T T F F T

T F F F T

F T T T T

F F T F F

25.

p � (q � r)p q r p � q q � r (p � q) � r p � (q � r) ⇔ (p � q) � r

T T T T T T T T

T T F T F F F T

T F T F F F F T

T F F F F F F T

F T T F T F F T

F T F F F F F T

F F T F F F F T

F F F F F F F T

23.

p q p Q q p � (p Q q)

T T T T

T F F F

F T T F

F F T F

21.

p �p �p � p

T F T

F T T

19.

p q �p �p Q q

T T F T

T F F T

F T T T

F F T F


CHAPTER 11 Review


1. F 2. T 3. T4. F 5. T


1. p � q 2. �p3. logically equivalent 4. hypotheses; conclusion5. zero; one

CHAPTER 11 Review Questions (p. 617)

1. Proposition


5. Proposition

7. Not a porposition

9. I go to the math learning center, or I complete my mathhomework.

3. The output is 1 when p and q are both 1.

5.

7.

9. (For Problem 1):

(For Problem 3):

(For Problem 5):

(For Problem 7):

11. The truth table and two possible diagrams for this circuit iseither

p q

1 1 1

1 0 0

0 1 0

0 0 1

pq

pq XOR

pq

p

q

r

XOR

p

q NOR

AND

p

q OR

OR

AND

or

p q

1 1 0

1 0 1

0 1 1

0 0 0

(a) and

(b)

13.

15. p � q ≡ �(�p �q)

(a)

(b)

pq ⊕ pr ⊕ q(�r) ≡ pqr ⊕ pq (�r) ⊕ pr ⊕ q(�r)≡ pr(q ⊕ 1) ⊕ (p ⊕ 1)[q(�r)]

17. Show pq ⊕ pr ⊕ q(�r) � pr ⊕ q(�r)pq ⊕ pr ⊕ q(�r) � pq(r ⊕ �r) ⊕ pr ⊕ q(�r)

� pqr ⊕ pq(�r) ⊕ pr ⊕ q(�r)� pqr ⊕ pr ⊕ pq(�r) ⊕ q(�r)� pr(q � 1) ⊕ q(�r)(p � 1)� pr(1) ⊕ q(�r)(1)� pr � q(�r)

p

q

NAND

NAND

NAND

pq NAND

p

pq XOR

pq XOR


11. If I go to the math learning center then I complete my mathhomework.

13. If I don’t go to the math learning center, then I don’t com-plete my math homework.

15. I don’t go to the math learning center, and I don’t completemy math homework.

17. Nobody is rich.

19. Either Danny is tall or Mary is not short.

21. (c)

23. (a)

25.

p q p � q �p (p � q) � �p

T T T F T

T F F F F

F T F T T

F F F T T

27.

p q p � q (p � q) � p

T T T T

T F T T

F T T F

F F F F

29. q Q p

31. p ⇔ q

33. Define the statements p: The temperature outside is below30°, q: I wear gloves.(a) p Q q(b) q Q p; If I wear gloves, then the temperature outside is

below 30°.(c) �q Q �p; If I do not wear gloves, then the temperature

outside is not below 30°.(d) �p Q �q; If the temperature is not below 30°, then I do

not wear gloves.

35. Define the statements p: Stu will work on the project, q: Juliehelps.(a) q Q p(b) p Q q; If Stu works on the project, then Julie helps.(c) �p Q �q; If Stu will not work on the project, then Julie

does not help.(d) �q Q �p; If Julie does not help, then Stu will not work

on the project.

37. Define the statements p: Kurt will go to the club, q: Jessicacomes to town.(a) q Q p(b) p Q q; If Kurt goes to the club, then Jessica comes to

town.(c) �p Q �q; If Kurt does not go to the club, then Jessica

does not come to town.(d) �q Q �p; If Jessica does not come to town, then Kurt

will not go to the club.

39. Define the statements p: Brian must come to the gym, q:Mike works out.(a) q Q p(b) p Q q; If Brian must come to the gym, then Mike works

out.(c) �p Q �q; If Brian must not come to the gym, then

Mike does not work out.(d) �q Q �p; If Mike does not work out, then Brian must

not come to the gym.

41. p: Patrick goes to practice; q: Patrick starts the game Show p Q q is equivalent to �p � q.

p q �p p Q q �p � q p Q q PQ (�� q)

T T F T T T

T F F F F T

F T T T T T

F F T T T T

43. See table in problem 41 above

45. Define the statements p: I paint the house, q: I go bowling.Assume the premises �p Q q and �q are true. Show p is true.Direct Proof: Since �p Q q is true, its contrapositive �q Q p is true. When �q Q p and �q are true then by theLaw of Detachment, p is true, I paint the house.Indirect Proof: Assume the conclusion p is false. By the Lawof Contradiction �p is true. Since �p Q q and �p are true,by the Law of Detachment, q is true, but this is acontradiction. q is false. So the assumption that p is false isincorrect. p is true, I paint the house.

47. Define the statements p: John is in town, q: Mark gets tickets,and r: We go to the game. Assume the premises p Q q,�r Q �q, and p are true.Direct Proof: Since �r Q �q is true, its contrapositive,q Q r is true. Since p Q q and q Q r are true, p Q r is true,p Q r is true by Law of Syllogism. Finally, since p Q r and p aretrue, by the Law of Detachment, r is true. We went to the game.

49. Define the statements p: I pay a finance charge, q: Mypayment is late, r: Colleen sends the mail.Assume the premises q Q p, r � q, and �r are true. Prove pis true.

APPENDIX A Review


Direct Proof: Since r � q is true provided at least one of itscomponents is true and r is false by the Law ofContradiction, q if true. By the Law of Detachment, since q Q p and q are true, so is p. I pay a late charge.

51. Define the statements p: Rob is a bad boy, q: Danny is crying,r: Laura is a good girl.Assume the premises p � q, r Q �p, and �q are true. Wewant to prove r (or �r).Direct Proof: p � q is true whenever at least one of its com-ponents is true. �q is true, and by the Law of Contradictionq is false. So p is true. Since r Q �p is true, its contraposi-tive, p Q �r, is true, and by the Law of Detachment �r istrue. Lara is not a good girl.

53. p �(pq)q

55.

57. (p ⊕ q)[�(pq)] � (p ⊕ q)(�p ⊕ �q)� p(�p) ⊕ p(�q) ⊕ q(�p) ⊕ q(�q)� 0 ⊕ p(�q) ⊕ q(�p) ⊕ 0� p(�q) ⊕ (�p)q

p

q

�(pq)

p�q

(p�q) [�(pq)]

p

q

Exercise Appendix A.1 (p. 635)

1. (a) 2 and 5 are natural numbers.(b) �6, 2, and 5 are integers.(c) �6, , �1.333 . . . , 2, and 5 are rational numbers.(d) π is an irrational number.(e) All the numbers are real numbers.

3. (a) 1 is a natural number.(b) 0 and 1 are integers.(c) All the numbers are rational numbers.(d) There are no irrational numbers in the set C.(e) All the numbers are real numbers.

5. (a) There are no natural numbers in the set E.(b) There are no integers in the set E.(c) There are no rational numbers in the set E.(d) All the numbers are irrational.(e) All the numbers are real numbers.

7. (a) 18.953 9. (a) 28.653(b) 18.952 (b) 28.653

11. (a) 0.063 13. (a) 9.999(b) 0.062 (b) 9.998

15. (a) 0.429 17. (a) 34.733(b) 0.428 (b) 34.733

19. 3 � 2 � 5 21. x � 2 � 3 � 4

23. 3y � 1 � 2 25. x � 2 � 6

27. � 6 29. 7 31. 6

33. 1 35. 37. �11133

x2

12

39. 11 41. �4 43. 1

45. 6 47. 49.

51. 53. 55.

57. � 59. 61.

63. 6x � 24 65. x2 � 4x

67. x2 � 6x � 8 69. x2 � x � 2

71. x2 � 10x � 16 73. x2 � 4

79. Subtraction is not commutative. Examples will vary.

81. Division is not associative. Examples will vary.

83. This is true by the symmetric property of real numbers.

85. All real numbers are either rational or irrational; no realnumber is both.

87. 0.99 . . . � 1


1.

3. � 5. � 7. � 9. �

11. � 13. x � 0 15. x � 2 17. x � 1

19.

21.

23. 1 25. 2 27. 6 29. 4

–1 0 2 4

–2 20

–2.5 –1 0

0.25

134

52

1522

160

1645

1336

7930

2320

445

27

Exercise Appendix B AN-59

31. �28 33. 35. 0 37. 1

39. 5 41. 1 43. 22 45. 2

47. (c) x � 0 49. (a) x � 3 51. none

53. (b) x � 1, (c) x � 0, (d) x � �1

55. {x � x � 5} 57. {x � x � �4}

59. C � 0° 61. C � 25°

63. x � 2 65. x � 6

67. x � �1 69. x � �4

71. x � �3,

73. x �1,

75. x 1,

77. x � �4,

79. A � l � w; {

81. C � � d;

83. A � x2;

85. V � r3;

87. V � x3;

89. (a) $6,000(b) $8,000

91. (a) �113 � 115� � ��2� � 2 � 5(b) �109 � 115� � ��6� � 6 � 5

93. (a) Yes, �2.999 � 3� � ��0.001� � 0.001 � 0.01.(b) No, �2.89 � 3� � ��0.11� � 0.11 � 0.01.

95. No, is larger by 0.000333. . . .

97. No.


1. 64 3. 5. 1

7. 4 9. 3 11. 2

13. 4 15. 17.14

164

18

13

{x � x � 0}, {V �V � 0}

{r � r � 0}, {V �V � 0} 43

{x � x � 0}, {A�A � 0} √34

{d �d � 0}, {C �C � 0}

l �l � 0}, {w � w � 0}, {A� A � 0}

–4 –2 20

–2 0 31

–1–3 0 2

–2–3–5 0 2

45 19. (a) 11.2116 21. (a) 8.8152

(b) 11.5873 (b) 8.8214(c) 11.6639 (c) 8.8244(d) 11.6648 (d) 8.8250

23. (a) 21.2166(b) 22.2167(c) 22.4404(d) 22.4592

25. 3 27. �1 29. 8

31. 33. 2 35. 3

37. x � log2 5 39. t � log1.1 10 41. 55.590

43. 1385.002 45. 1499.364 47. 12,432.323

49. 2074.642


1. 1, 2, 3, 4, 5 3. , , , ,

5. 1, �4, 9, �16, 25 7. , , , ,

9. , , , , 11. , , , ,

13. 1, 3, 5, 7, 9 15. �2, �1, 1, 4, 8

17. 5, 10, 20, 40, 80 19. 3, 3, , ,

21. 1, 2, 2, 4, 8

23. A, A � d, A � 2d, A � 3d, A � 4d

25. , , , ,

27. (a) a1 � 2, r � 2 29. (a) a1 � , r �

(b) 2, 4, 8, 16 (b) , , ,

(c) 30 (c)

31. (a) a1 � , r � 2 33. (a) a1 � , r �

(b) , , 1, 2 (b) , , 2, 2

(c) (c) 2 � 3 �

35. (a) a1 � , r �

(b) , , ,

(c) 6516

2716

98

34

12

32

12

√3 4√3 2154

√3 2√3 4√3 212

14

√3 2√3 214

�4516

� 316�3

8�34�3

2

12�3

2

√2 � √2 � √2 � √2 � √2

√2 � √2 � √2 � √2√2 � √2 � √2√2 � √2√2

18

12

32

5e5

4e4

3e3

2e2

1e� 1

421

30� 120

112�1

6

861

841

27

25

12

56

45

34

23

12

13

APPENDIX B Using Lindo to Solve Linear Programming Problems

Exercise Appendix B (p. 659)

1. Maximum of P � 24 when x1 � 0, x2 � 12, and x3 � 0.




9. Maximum of P � 15.2 when x1 � 1.6, x2 � 4.8, and x3 � 2.4.


13. No maximum value.

AN-60 Appendix Answers

APPENDIX C Graphing Utilities

Exercise Appendix C.1 (p. 663)

1. (�1, 2) quadrant II 3. (3, 1) quadrant I

5. X min � �6 7. X min � �6X max � 6 X max � 6X scl � 2 X scl � 2Y min � �4 Y min � �1Y max � 4 Y max � 3Y scl � 2 Y scl � 1

9. X min � 3 11. X min � �12X max � 9 X max � 6X scl � 1 X scl � 1Y min � 2 Y min � �4Y max � 10 Y max � 8Y scl � 2 Y scl � 1


17. Maximum of P � 42 when x1 � 1, x2 � 10, x3 � 0, and x4 � 0.


21. Maximum of P � 50 when x1 � 0, x2 � 15, x3 � 5, and x4 � 0.

23. Minimum of P � 76.25 when x1 � 6.25, x2 � 0, x3 � 0, x4 �20, x5 � 0, x6 � 50, and x7 � 0.

13. X min � �30 15. X min � �10X max � 50 X max � 110X scl � 10 X scl � 10Y min � �100 Y min � �20Y max � 50 Y max � 180Y scl � 10 Y scl � 20

Exercise Appendix C.2

1. (a) (b) (c) (d)

3. (a) (b) (c) (d)

5. (a) (b) (c) (d)

– 5 5

20

– 20

–10 10

8

– 8

–10 10

8

– 8

– 5 5

4

– 4

– 5 5

20

– 20

–10 10

8

– 8

–10 10

8

– 8

– 5 5

4

– 4

– 5 5

20

– 20

–10 10

8

– 8

–10 10

8

– 8

– 5 5

4

– 4

Exercise Appendix C-2 AN-61

7. (a) (b) (c) (d)

9. (a) (b) (c) (d)

11. (a) (b) (c) (d)

13. (a) (b) (c) (d)

15. (a) (b) (c) (d)

17. 19. 21. 23.

– 5 5

20

– 20

–10 10

8

– 8

–10 10

8

– 8

– 5 5

4

– 4

– 5 5

20

– 20

–10 10

8

– 8

–10 10

8

– 8

– 5 5

4

– 4

– 5 5

20

– 20

–10 10

8

– 8

–10 10

8

– 8

– 5 5

4

– 4

– 5 5

20

– 20

–10 10

8

– 8

–10 10

8

– 8

5 5

4

– 4

– 5 5

20

– 20

–10 10

8

– 8

–10 10

8

– 8

– 5 5

4

– 4


Exercise Appendix C.3

1. Yes 3. Yes 5. No 7. Yes

9. Y min � 1Y max � 9Y scl � 1

25. 27.

29. 31.

Documents

Answers to Odd-Numbered Problems · AN-6 Answers to Odd-Numbered Problems (f) 11. (a) (b)y x 115 (c) (d) Window: X min 0; X max 100 Y min 1; Y max 120 y x 0 0 20406080100 20 40 60