precalculus with limits, answers to section 7.1 1

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d.Precalculus with Limits, Answers to Section 7.1 1

Chapter 7Section 7.1 (page 503)

Vocabulary Check (page 503)1. system of equations 2. solution3. solving 4. substitution5. point of intersection 6. break-even

1. (a) No (b) No (c) No (d) Yes2. (a) Yes (b) No (c) No (d) Yes3. (a) No (b) Yes (c) No (d) No4. (a) No (b) Yes (c) Yes (d) No5. 6. 7.

8.9. 10.

11. 12.13. 14.15. 16. 17. 18.

19. 20. 21. 22.23. No solution 24. No solution 25.26. 27. No solution 28.29. 30. 31. 32.33. 34. 35.36. 37. 38.39. No solution 40. No solution 41.42.

43. 44.

45. 46.

47. 48.

49. 50. No solution 51.

52. 53. No solution 54. No solution

55. 56.57. 58.

59. 60. 61. 192 units

62. 3133 units 63. (a) 781 units (b) 3708 units64. (a) 3760 items (b) 10,151 items65. (a) 8 weeks

(b)

66. (a) 5 weeks(b)

67. More than $11,666.67

68.

69. (a)

(b)

Decreases; Interest is fixed.(c) $5000

70. (a)

(b) 24.7 inches(c) Doyle Log Rule

00

40

V1

V2

1500

0 10,00012,000

27,000

� x0.06x

�

�

y0.085y

�

�

25,0002,000

�99.99, 2.85�

00

150

10

�1, 0�, �5, 2�� 12, 2�, ��4, �1

4��0, �1�, �2, 1�, ��1, �5��1, 0�, �0, 1�, �1, 0��1.96, 0.14�, �1.06, 2.88��0.287, 1.751�

�3, 4�, �5, 0��2, 0�, � 29

10, 2110��1, 2�

�0, �2�, �±1.32, 1.50��0, �13�, �±12, 5�

−6

−4

6

4

−16

−24 24

16

�5.31, �0.54��4, 2�

−1

−6

4

14

−2 10

−3

5

��0.49, �6.53��0, 1�

−7

−10

80

−2

6−6

6

�3, �4��3, 4�,�4, 3�, ��4, 3��2, 2��4, �1

2��15, 7��3, 1�,�1, 4�, �4, 7��3, 6��3, 0�,�2, 2�, �4, 0��5, 3�� 5

2, 32��2, �2��4, 3��0, 0��12, 6��0, 0�,��2, 4�, �0, 0�

�20817 , 88

17�� 203 , 40

3 ��2, 52��1, 1��4

3, 43�� 12, 3��3, 2��5, 5�

�0, 4�, �1, 2�, �2, 0��0, 1�, �1, �1�, �3, 1��0, 2�, �1, 0�, ��1, 0��0, 0�, �2, �4��0, 0�, �2, �2�, ��2, 2��0, �5�, �4, 3�

��3, 2 � 3�3 ��0, 2�,��3, 2 � 3�3 �,�2, 6�, ��1, 3��1, 3��2, 2�

1 2 3 4

336 312 228 264

42 60 78 9624 � 18x

360 � 24x

5 6 7 8

240 216 192 168

114 132 150 16824 � 18x

360 � 24x

1 2 3 4 5

125 150 175 200 225

425 375 325 275 225�50x � 475

25x � 100

333202CB07_AN.qxd 4/13/06 5:30 PM

Precalculus with Limits, Answers to Section 7.1 2

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d.

(Continued)

71. (a) Solar:Wind:

(b)

(c) Point of intersection: Consumption ofsolar and wind energy are equal at this point in time inthe year 2000.

(d)(e) The results are the same, but due to the given parame-

ters, is not of significance.(f) Answers will vary.

72. (a) Alabama:Colorado:

(b) Point of intersection: Colorado’spopulation exceeded Alabama’s just after this point.

(c) so 73.74.75. 76.77.78.79. False. To solve a system of equations by substitution, you

can solve for either variable in one of the two equations andthen back-substitute.

80. False. The system can have at most four solutions becausea parabola and a circle can intersect at most four times.

81. 1. Solve one of the equations for one variable in terms ofthe other.

2. Substitute the expression found in Step 1 into the otherequation to obtain an equation in one variable.

3. Solve the equation obtained in Step 2.4. Back-substitute the value obtained in Step 3 into the

expression obtained in Step 1 to find the value of theother variable.

5. Check that the solution satisfies each of the originalequations.

82. For a linear system, the result will be a contradictoryequation such as , where is a nonzero real num-ber. For a nonlinear system, there may be an equation withimaginary solutions.

83. (a) (b) (c)84. (a)

(b) There are three points of intersection when b is even.

85. 86.87. 88.89. 90.91. Domain: All real numbers except

Horizontal asymptote: Vertical asymptote:

92. Domain: All real numbers except

Horizontal asymptote:

Vertical asymptote: 93. Domain: All real numbers except

Horizontal asymptote: Vertical asymptotes:

94. Domain: All real numbers except Horizontal asymptote: Vertical asymptote: x � 0

y � 3x � 0x

x � ±4y � 1

x � ±4xx � �

23

y �23

x � �23x

x � 6y � 0

x � 6x45x � 29y � 127 � 030x � 17y � 18 � 0

x � 4 � 0y � 3 � 0

4x � 13y � 38 � 02x � 7y � 45 � 0

−6

−2

6

6

−6

−2

6

6

b � 4b � 3

−6

−2

6

6

−6

−2

6

6

b � 2b � 1y � x � 2y � 0y � 2x

N0 � N

�2 inches � �2 inches � 2 inches8 kilometers � 12 kilometers

42 feet � 63 feet9 inches � 12 inches 60 centimeters � 80 centimeters6 meters � 9 meters

t � 11.9317.4t � 4273.2 � 84.9t � 3467.9,

�11.93, 4480.79�.

94000

13

4800

84.9t � 3467.917.4t � 4273.2

t � 135.47

t � 10.3, 135.47

�10.3, 66.01�.

80

13

150

16.371t � 102.70.1429t2 � 4.46t � 96.8

333202CB07_AN.qxd 4/13/06 5:30 PM

Precalculus with Limits, Answers to Section 7.2 3C

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Section 7.2 (page 515)

Vocabulary Check (page 515)1. elimination 2. equivalent3. consistent; inconsistent 4. equilibrium point

1. 2.

3. 4.

5. No solution 6. No solution

7. 8.

9. 10.

11. 12. 13. 14.

15. 16. 17. 18.

19. No solution 20. Infinitely many solutions:

21. 22. Infinitely many solutions:

23. Infinitely many solutions:

24. Infinitely many solutions:

25. 26. 27.

28. 29. 30.31. b; one solution; consistent32. a; infinitely many solutions; consistent33. c; one solution; consistent34. d; no solutions; inconsistent35. 36. 37. 38.39. 40. 41.42. 43. 550 miles per hour, 50 miles per hour44. First plane: 880 kilometers per hour

Second plane: 960 kilometers per hour45. 46. 47.48.49. Cheeseburger: 310 calories; fries: 230 calories50. Apple juice: 103 milligrams; orange juice: 82 milligrams51. (a)

(b) (c) 20% solution:

50% solution:

Decreases

52. (a)

(b) (c) 87 octane: 300 gallons;92 octane: 200 gallons

Decreases53. $6000 54. $20,000 55. 400 adult, 1035 student

00

500

500

� x �

87x �

y �

92y �

500

44,500

313 liters

623 liters

−6 18

−4

12

� x �

0.2x �

y � 10

0.5y � 3

�250,000, 350��2,000,000, 100��500, 75��80, 10�

�3, 2��43

6 , 256 ��23, 61��6, �3�

�1, 3��2, �1��2, 5��4, 1�

�7, 1��5, �2��149 , 20

9 �� 6

35, 4335 ��101, 96�� 90

31, �6731 �

�a, 34 �78a�

�a, �12 �

56a�

�a, 4 � 4a�� 185 , 35�

�a, 18 �34a�

��17, 5��127 , 18

7 ��56, 56��4, �1�

�5, 1��3, 4��3, 75�� 52, 34�

x

5x + 3y = −18

2x − 6y = 1

2−4

−2

−6

4

2

y

x

−4

−3

−2

3

4

y

−1 2 3 4−2−3−4

9x + 3y = 1

3x − 6y = 5

��3512, �41

36�� 13, �2

3�

x

6

8

−4

−6

−8

42−4−6−8 86

y −3x + y = 5

9x − 3y = −15

x

y

−1 2 3 4 5−2

−2

1

2

3

4

−3

−6x + 4y = −10

3x − 2y = 5

�a, 3a � 5��a, 32 a �52�

x

6x + 4y = 14

−2 2

−2

−4

4

y

3x + 2y = 3

x

y

−1

−2

1

4

−4

2 3 4−2−4

−2x + 2y = 5

x − y = 2

x

2x − y = 3

4x + 3y = 21

−2

2

4

6

42

y

x

2

3

4

y

−1 2 3 4−2

−4

−3

−2

−3−4

3x + 2y = 1x + y = 0

�3, 3��1, �1�

x−2

−2

−4−6

4

y

−x + 2y = 4x + 3y = 1

x65421−1−2

−4

−3

1

2

3

4

2x + y = 5

x − y = 1y

��2, 1��2, 1�

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d.

(Continued)

56. Before noon: 81 jackets; After noon: 133 jackets57. 58.59. 60.

61. 62.63. (a) (b) 41.4 bushels per acre64. (a) and (b)

(c)

(d) $92.54 (e) 200465. False. Two lines that coincide have infinitely many points

of intersection.66. False. Solving a system of equations algebraically will

always give an exact solution.67. No. Two lines will intersect only once or will coincide, and if

they coincide the system will have infinitely many solutions.68. (a) (b)

69. (39, 600, 398). It is necessary to change the scale on theaxes to see the point of intersection.

70. It is necessary to change the scale on the axesto see the point of intersection.

71. 72.

73. 74.

75. 76.

77. 78. All real numbers x

79. 80.

81. 82. 83.


87. Answers will vary.

�32, 3

10�log6 4�3x

log9 12x

ln x

�x � 3�5ln 6x

x

420−2−4−6−8x

72

−6 −5 −4 −3 −2 −1 0 1 2 3 4

x > 0x < �4,�5 < x < 72

x

3210−1−2−3x

1815129630

−2

−3

�2 < x < 18

x

7654321

163

43

x

3210

1916

−1

43 ≤ x < 16

3x ≤ 1916

x

43210

−5−6−7−8−9

x

223

−

x > 1x ≤ �223

k � �2k � �4

�300, 315�.

� x � y � 3

2x � 2y � 6�x � y � 10

x � y � 20

y � 3.6t � 49.343y � 14x � 19

y �12 x �

34y � �2x � 4

y � �0.58x � 5.4y � 0.32x � 4.1y � 0.22x � 1.9y � 0.97x � 2.1

Year 1995 1996 1997 1998

y $67.34 $70.94 $74.54 $78.14

Year 1999 2000 2001

y $81.74 $85.34 $88.94

333202CB07_AN.qxd 4/13/06 5:30 PM


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Vocabulary Check (page 527)1. row-echelon 2. ordered triple3. Gaussian 4. row operation5. nonsquare 6. position

1. (a) No (b) No (c) No (d) Yes

2. (a) No (b) Yes (c) No (d) No

3. (a) No (b) No (c) Yes (d) No

4. (a) Yes (b) No (c) No (d) Yes

5. 6. 7.

8. 9. 10.

11. 12.

First step in putting the Eliminated the x-termsystem in row-echelon form from the 3rd equation.

13. 14. 15.

16. 17. 18.

19. No solution 20. No solution 21.

22. 23.

24. 25.

26. 27.

28. 29.

30. 31.


35. 36. 37.

38. 39.

40. 41.

42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 6 touchdowns, 6 extra-point kicks, and 1 field goal

52. 17 two-point baskets, 7 three-point baskets,15 one-point free throws

53. $300,000 at 8% 54. $625,000 at 8%$400,000 at 9% $50,000 at 9%$75,000 at 10% $125,000 at 10%

55. in certificates of deposit

in municipal bondsin blue-chip stocks

s in growth stocks56. in certificates of deposit,

in municipal bonds,

in blue-chip stocks,in growth stocks

57. Brand 58. 20 liters of spray XBrand 18 liters of spray YBrand 16 liters of spray Z

59. 60.

61. 62.

63. (a) Not possible(b) No gallons of 10%, 6 gallons of 15%, 6 gallons of 25%(c) 4 gallons of 10%, No gallons of 15%, 8 gallons of 25%

64. (a) 1 liter of 10%, 7 liters of 20%, 2 liters of 50%

(b) No liters of 10%, liters of 20%, liters of 50%

(c) liters of 10%, No liters of 20%, liters of 50%

65.

66. (a)

(b)

The system is stable.a � 0 feet per second squaredt2 � 64 poundst1 � 128 poundsa � �16 feet per second squaredt2 � 48 poundst1 � 96 pounds

I3 � 1I2 � 2,I1 � 1,

33461

4

12381

3

Pop � 9Newspaper � 20 adsDance � 5Radio � 10 adsRock � 18Television � 30 ads

Irises � 3French Roast � 4 lbLilies � 1Hazelnut � 4 lbRoses � 8Vanilla � 2 lb

Z � 9 lbY � 9 lbX � 4 lb

s125,000 � s

�31,250 �12s

406,250 �12s

125,000 � s125,000 �

12s

250,000 �12s

−2

−3

4

1

−12

−2

6

10

x2 � y 2 � 3x � 2y � 0x2 � y 2 � 6x � 8y � 0

−6

−1

6

7

−3 6

−3

3

x2 � y 2 � 6y � 0x2 � y 2 � 4x � 0

−4 8

4

−4

−6

−2

12

10

y � �2x 2 � 5xy � x2 � 6x � 8

−5

−3

7

5

−4 8

−3

5

y � �x 2 � 2x � 3y �12x 2 � 2x

s � �16t 2 � 16t � 132

s � �16t 2 � 32t � 500s � �16t 2 � 64t

s � �16t 2 � 144�15a �

15, �3

5a �25, a�

�9a, �35a, 67a��0, 0, 0��0, 0, 0�

�3, 72, 12��1, 0, 3, 2�

�1, 1, 1, 1��38a �

14, �3

4a �52, a�

��32a �

12, �2

3a � 1, a��5a � 3, �a � 5, a��2a, 21a � 2, 8a��2a � 5, �7a � 14, a�

��a � 3, a � 1, a��12a �

52, 4a � 1, a�

��3a � 10, 5a � 7, a�� 310, 25, 0�

��12, 1, 32�

�1, 12, �3��5, �2, 0��5, �3, 3��4, 8, 5��5

3, 13, 1��1, 2, 3�

�x � 2y � 3z �

�x � 3y � 5z �

4y � 9z �

5

4

�10�

x

2x

� 2y

y

� 3z � 5

� 2z � 9

� 3z � 0

��2, �103 , �4�� 1

2, �2, 2��17, �11, �3��3, 10, 2��2, �3, �2��1, �2, 4�

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d.

(Continued)

67. 68.

69. 70.

71. (a)

(b) (c)

The values are the same.

(d) 24.25% (e) 156 females

72. (a)

(b) (c) 453 feet

73.

74.

75. 76.

77. or 78.

79. False. Equation 2 does not have a leading coefficient of 1.

80. True. If a system of three linear equations is inconsistent,then it has no points common to all three equations.

81. No. Answers will vary.

82. There will be a row representing a contradictory equationsuch as where is a nonzero real number.

83.

84.

85.

86.

87. 6.375 88. 150% 89. 80,000 90. 275

91. 92. 93.

94. 95. 96.

97. (a)

(b)

98. (a)

(b)

99. (a)

(b)

100. (a)

(b)

x8642−2−4

20

y

�12, 13, 5

x1−2−3−5 2 4

30

20

10

−30

−40

−50

−60

y

�4, �32, 3

x

36

30

24

18

−631−1−3

y

±2, 0

x1−1−2−3−5 2 4

25

20

15

−10

−15

−20

y

�4, 0, 3

1751241 �

201241i7

2 �72i11 � 2i

22 � 3i�7 � 3i11 � i

� 4x � y � 2z � 124y � 2z � 2

�2x � y � z � 0�

2x � y � 3z �

�6x � 4y � z �

�4x � 2y � 3z �

�28

18

19

�x � 2y � 4z � 9

y � 2z � 3

x � 4z � �4�

x � 2y � 4z �

�x � 4y � 8z �

x � 6y � 4z �

�5

13

7

� 2x � y � z � �9�x � 2y � 2z � 3

�3x � y � 2z � 11�

x � y � z �

�2x � y � 3z �

x � 4y � z �

�6

15

�14

�x � y � z � 5

x � 2z � 0

2y � z � 0�

3x � y � z � 9

x � 2y � z � 0

�x � y � 3z � 1

N0 � N,

� � �51� � 0� � 1

y � 50y � 0y �12

x � 25x � 0x � ±�2� 2

� � �4� � �5

y � 2y � 5

x � 2x � 5

Safeties � 1Field goals � 3;

Extra-point kicks � 9;Touchdowns � 9;

Extra-point kicks � 5Two-point conversions � 1;

Field goals � 2;Touchdowns � 8;

Speed(in miles per hour)

Stop

ping

dis

tanc

e(i

n fe

et)

x10 7050403020

50100150200250300350400450

60

y

y � 0.165x2 � 6.55x � 103

750

100

175

y � �0.0075x2 � 1.3x � 20

y �37 x 2 �

65 x �

2635y � �

524 x2 �

310x �

416

y � �54 x 2 �

920 x �

19920y � x2 � x

x 100 120 140

y 75 68 55

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(Continued)

101.

102.

103.

104.

105. 106. 107. Answers will vary.�12, 0��40, 40�

1 2 3 4−1

18

12

6

x

−6

y

x654321−1−2−3

7

6

5

4

2

2

1

−2

y

x4321−1−2−3−4

12

10

8

6

4

2

−4

−6

y

x64321−1−2−3

12

10

8

6

4

2

−4

−6

y

0 2 4 5

�1�4�4.938�4.996�5y

�2x

x 0 1 2

y 11.625 2.25 �3.6�3�1.5

�1�2

0 1 2

5.793 4.671 4 3.598 3.358y

�1�2x

x 0 1 2

y 28.918 18.25 12.548 9.5 7

12�

12

333202CB07_AN.qxd 4/13/06 5:30 PM


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d.


Vocabulary Check (page 539)1. partial fraction decomposition 2. improper3. linear; quadratic; irreducible 4. basic equation

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

5. 6.

7. 8.

9.

10.

11. 12.

13.

14.

15. 16.

17. 18. 19.

20. 21.

22. 23.

24. 25.

26. 27.

28. 29.

30. 31.

32.

33.

34.

35.

36. 37.

38. 39.

40. 41.

42.

43.

44.

45. 46.

47. 48.

49.

50.

51.

52.

53. (a)

(b)

(c) The vertical asymptotes are the same.

54. (a)

(b)


−1 1 2 3 4

1

2

4

x

y = 2x

y = 4x2 + 1

y

–1–2–3–4 1 2 3 4

–4

1

2

3

4

x

y

y �2x, y �

4x2 � 1

y �2�x � 1�2

x�x2 � 1�

2x

�4

x2 � 1

−6 2 8 10

−8

2

8

x

y = 3x

y = 3x

y = 2x − 4

−

y = 2x − 4

−

y

−6 −4 2 8 10

−8

2

4

6

8

x

y

y �3x, y � �

2x � 4

y �x � 12

x�x � 4�

3

x�

2

x � 4

x � 1 �1

x � 2�

1

x � 1

2x �12�

3x � 4

�1

x � 2�

1

21

x � 2�

1

�x � 2�2�

1

x � 2�

1

�x � 2�2

1

x2 � 2�

x

�x2 � 2�2

1

2�1

x�

5

x � 1�

3

�x � 1�22

x�

1

x2�

2

x � 1

2

x�

4

x � 1�

3

x � 1

3

2x � 1�

2

x � 1

2x � 3 �6

2x � 1�

4�2x � 1�2 �

1�2x � 1�3

x � 3 �6

x � 1�

4

�x � 1�2�

1

�x � 1�3

x � 1 �15�

27x � 4

�3

x � 1�2x � 7 �

17x � 2

�1

x � 11 �

5x � 6x2 � x � 6

1 �2x � 1

x2 � x � 12

x � 1�

x � 1

x 2 � 2x � 3

1

x � 1�

2

x2 � 2x � 3

1

x�

x � 1

x2 � 1

1

8 �1

2x � 1�

1

2x � 1�

4x

4x2 � 1�

2

x 2 � 4�

x

�x 2 � 4�2

1

6 �2

x2 � 2�

1

x � 2�

1

x � 2�

15�

9x � 3

�1

x � 2�

10x � 2�

�1

x � 1�

x � 2x2 � 2

1

3�1

x � 1�

x � 1

x 2 � x � 1��

1

x�

2x

x2 � 1

2

x�

1

x 2�

2

x � 1�

7

�x � 1� 2

3

x � 3�

9

�x � 3�2

2

x � 1�

1

�x � 1�2

3

x�

1

x2�

1

x � 1

1

2�3

x � 4�

1

x��

3

x�

1

x � 2�

5

x � 21

x � 3, x � �1

1

x � 1�

1

x � 2

1

x � 2�

1

x � 3

1

x�

2

2x � 1

1

x � 3�

1

x

1

x�

1

x � 1

1

6�1

2x � 3�

1

2x � 3�12�

1x � 1

�1

x � 1�

Ax

�Bx2 �

C3x � 1

�D

�3x � 1�2

Ax

�Bx � Cx2 � 1

�Dx � E

�x2 � 1�2

A2x

�Bx � Cx2 � 4

A

x�

Bx � C

x2 � 10

Ax � 2

�B

�x � 2�2 �C

�x � 2�3 �D

�x � 2�4

Ax � 5

�B

�x � 5�2 �C

�x � 5�3

Ax

�Bx2 �

C4x � 11

A

x�

B

x2�

C

x � 10

A

x � 3�

B

x � 1

A

x�

B

x � 14

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55. (a)

(b)


56. (a)

(b)


57. (a)

(b) Ymax

Ymin

(c) (d) Maximum: Minimum:

58. Answers will vary. Sample answer: You can substitute anyconvenient values of that will help determine constants.You can also find your basic equation, expand it, thenequate coefficients of like terms.

59. False. The partial fraction decomposition is

60. False. The expression is an improper rational expression,so you must first divide before applying partial fractiondecomposition.

61. 62.

63. 64.

65. 66.

67. 68.

69. 70.

x−2−4

2

64

−8

−10

−4

8

4

6

−6

y

x

5

5−10−15−20 10 15 20

y

x−1−2−3

1

32

−2

−3

−4

2

y

x

3

4

5

−1 1 2 4 5−2−3−1

−2

−3

y

x−2−4−6 2

−4

−6

−8

−10

−12

−14

−16

4 6 8 10 12

y

x

2

4

6

8

−2 2 4 8 10−2

−4

y

1a � 1�

1x � 1

�1

a � x�1a �

1y

�1

a � y�

1a�

1x

�1

x � a�12a �

1a � x

�1

a � x�

Ax � 10

�B

x � 10�

C�x � 10�2.

x

266.7�F400�F

0

−100

1

Ymax

Ymin

1000

� � �200011 � 7x�

� � 20007 � 4x�

2000

7 � 4x�

2000

11 � 7x, 0 < x ≤ 1

−2−4 2 4 6 8 10 12

−4

10

12

x

y =

y = 5x2 − 10x + 26

3x2

y

−2−4 2 4 6 8 10 12

−4

10

12

x

y

y �3x2, y �

5x2 � 10x � 26

y �2�4x2 � 15x � 39�x2�x2 � 10x � 26�

3x2 �

5x2 � 10x � 26

−4 2 4 6 8

−8

−6

−4

6

8

x

y = 5x + 3

y = 5x + 3

y = 3x − 3

y = 3x − 3

y

−4 4 6 8

−8

−6

−4

4

6

8

x

y

y �3

x � 3, y �

5x � 3

y �2�4x � 3�

x2 � 9

3

x � 3�

5

x � 3

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Vocabulary Check (page 548)1. solution 2. graph 3. linear4. solution 5. consumer surplus

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.−8

−10

8

2

−50

4

6

−6

−4

6

4

−9

−9

9

3

−18

−22

18

2

−3

−2

3

2

−6

−1

9

9

−6

−4

6

4

−12

−8 4

4

−2

−8 1

4

−2

−9 9

10

−2

0 6

2

x321−1−2−3

3

2

1

−2

−3

−5

y

−3 −2 −1 1 2 3

−3

−2

2

3

x

y

y

x−1−2−3−4 1 2 3 4

−1

1

2

3

4

5

6

7

−5 −4 2 3

−2

1

2

3

4

6

x

y

−6 −4 2 4

−8

−6

2

x

y

−4 −3 −2 −1 1

−2

1

3

4

x

y

−3 −2 −1 1 3 4 5

−4

−3

−2

1

2

x

y

−2 −1 1 2 3 4

−2

1

2

3

4

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

4

x

y

1−1−2−3 2 3

−2

1

2

3

4

x

y

−1 1 2 3 5

−3

−2

−1

1

2

3

x

y

−1 1 3 4 5

−3

−2

−1

1

2

3

x

y

−1 1 2 3 4 5

−3

−2

−1

1

2

3

x

yy

x−1−2−3 1 2 3

−1

−2

−3

1

3

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27.

28.

29.30.31. (a) No (b) No (c) Yes (d) Yes32. (a) Yes (b) No (c) No (d) No33. (a) Yes (b) No (c) Yes (d) Yes34. (a) No (b) No (c) No (d) Yes35. 36.

37. 38.

39. 40.

No solution

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

55. 56. 57.

58. x2 � y2 > 4

�y ≥ 4 � x

y ≥ 2 �14 x

x ≥ 0, y ≥ 0�

y < 6 � 2x

y ≥ x � 3

x ≥ 1�

y ≤ 4 � x

x ≥ 0

y ≥ 0

−3

−1

3

3

−2 7

−1

5

−3

−2

3

2

−6 6

−3

5

−4

−3

8

5

−5

−1

7

7

−3 −2 1

−1

1

3

x

(0, 0)

(−3, 3)

y

x54321

4

3

2

1

−3

−4

(4, 4)

y

(−1, −1)

x2−2−6 4 6

−6

−4

2

4

6

(3, 4)

(−3, −4)

y

−4 −2 2 4

−4

−2

2

4

x

y

1 2 3 4

−2

−1

1

2

x

(4, 2)

(1, −1)

y

−1 1 2 3 4 5

−3

−2

1

2

3

x

(4, 2)

(1, −1)

y

2 4 6

–2

4

6

x

(0, 3)

y

( (

−3 −1 1 3 4

−3

−2

1

3

5

x(−2, 0)

,

y

109

79

x

(6, 6)

(1, 0)

y

2 4 6

2

6

−2 −1 2 3 4

−2

−1

1

4

x

y

x4−2−4

(2, 1)

−2

−4

−6 (2, −6)

, 1 22( (

, 1 − 22( (

y

x4321−3−4

6

4

3

2

1

(−1, 4)

(−1, 0) 5, 0( (

y

1 3

1

2

3

x(0, 0) (2, 0)

(0, 3)

y

−2 1 2

−1

2

3

x(−1, 0) (1, 0)

(0, 1)

y

x2 � y2 ≤ 9y ≥ �

23 x � 2

y ≥ x2 � 4

y ≤ 12 x � 2

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(Continued)

59. 60. 61.

62. 63.

64.

65. (a) (b) Consumer surplus: $1600Producer surplus: $400

66. (a) (b) Consumer surplus: $6250Producer surplus: $12,500

67. (a)

(b) Consumer surplus: $40,000,000Producer surplus: $20,000,000

68. (a)


69.

70.

71.

72.

73.

74.

1 2 4 5

1

2

5

6

y

x

�6x

3x

x

�

�

4y ≥6y ≥

≥y ≥

15

16

0

0

100

120

80

60

40

20

x20 40 60 80 100 120

y

�55xx

� 70y ≤≥

y ≥

75005040

500

1500

2500

x500 1500 2500 3500 4500

3500

4500

y

�x

30x

x

x

�

�

y ≤20y ≥

≤≥

y ≥

3000

75,000

2000

0

0

10,000 15,000

10,000

15,000

x

y

�x

x

� y

y

y

≤≥≥≥

20,000

2x

5,000

5,000

8 12 16 20 24

4

8

12

16

20

24

x

y

�x

8xx

� 12y

y

≥ 2y

≤ 200

≥ 4

≥ 2

2 4 6 8 10

2

4

6

10

12

x

y

�x

43 x

x

�32 y

�32 y

y

≤ 12

≤ 15

≥ 0

≥ 0

x

600

500

400

300

200

100

200,000 400,000

(250,000, 350)

p = 400 − 0.0002x

p = 225 + 0.0005x

Consumer SurplusProducer Surplus

p

x1,000,000 2,000,000

80

100

120

140

160

p = 80 + 0.00001x

p = 140 − 0.00002x

(2,000,000, 100)


p

x

200

150

100

50

200 400 600

(500, 75)

p = 25 + 0.1x

p = 100 − 0.05x


p

x10 60 70 80504020 30

10

20

30

40

50

(80, 10)


p

p = 50 − 0.5x

p = 0.125x

�y ≤ x � 1

y ≤ �x � 1

y ≥ 0

� y ≤ 32 x

y ≤ �x � 5y ≥ 0�

4x � y ≥ 0 4x � y ≤ 16

0 ≤ y ≤ 4

�2 ≤ x ≤ 5

1 ≤ y ≤ 7�x2 � y2 ≤ 16

x ≤ y

x ≥ 0 �

x2 � y2 ≤ 16

x ≥ 0

y ≥ 0

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75. (a) (b)

(c) Answers will vary.

76. (a)

(b)

(c) Answers will vary.77. (a)

(b)

(c)

78. (a) (b)

79. True. The figure is a rectangle with a length of 9 units anda width of 11 units.

80. False. The graph shows the solution of the system

81. The graph is a half-line on the real number line; on therectangular coordinate system, the graph is a half-plane.

82. Test a point on either side.

83. (a) (b)

(c) The line is an asymptote to the boundary. The largerthe circles, the closer the radii can be and the constraintwill still be satisfied.

84. (a) The boundary would be included in the solution.

(b) The solution would be the half-plane on the oppositeside of the boundary.

85. d 86. b 87. c 88. a

89. 90.

91. 92.

93. 94.

95. (a)

(b)

(c) The quadratic model is the best fit for the data.(d) $48.66

y1

y2

y3

530

18

60

y3 � 27�1.05t�

y2 � �0.241t2 � 7.23t � 3.4

y1 � 2.17t � 22.5

60x � 35y � 113 � 0x � y � 1.8 � 0

2x � y � 1 � 028x � 17y � 13 � 0

x � 11y � 8 � 05x � 3y � 8 � 0

−6 6

−4

4

�y2 � x2 ≥y >x >

10x0

� y < 6�4x � 9y < 6

3x � y2 ≥ 2.

10 20 30 40 50 60

10

20

30

50

60

x

y

�2x

x

�

xy ≥y ≥

≥y ≥

500

125

0

0

Total retail sales �h2

�a � b� � $821.3 billion

80

14

225

y � 19.17t � 46.61

y

x25 50 75 100

25

50

75

100

125

150

175

y ≥ 0.5�220 � x�y ≤ 0.75�220 � x�x ≥ 20x ≤ 70

30

30

x

y

�20x

15x

10x

x

�

�

�

10y

10y

20y

y

≥ 300

≥ 150

≥ 200

≥ 0

≥ 0

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Vocabulary Check (page 558)1. optimization 2. linear programming3. objective 4. constraints; feasible solutions5. vertex

1. Minimum at 0 2. Minimum at 0

Maximum at 20 Maximum at 32






Maximum at Maximum at 11

9. Minimum at 0

Maximum at

10. Minimum at 11,250

Maximum at 28,000

11. Minimum at

Maximum at any point on the line segment connectingand

12. Minimum at 6750

Maximum at 16,000

13. 14.

Minimum at Minimum at Maximum at Maximum at 15. 16.

Minimum at Minimum at Maximum at Maximum at 17. 18.

Minimum at Minimum at No maximum Maximum at 19. 20.

Minimum at Minimum at No maximum Maximum at 21. 22.

Minimum at Minimum at any point onMaximum at the line segment connecting

and Maximum at

23. 24.

Minimum at Minimum at any point onMaximum at the line segment connecting

and Maximum at

25. Maximum at 26. Maximum at

27. Maximum at

28. Maximum at any point on the line segment connectingand �5, 0�: 15�3, 6�

�0, 10�: 10

�5, 0�: 25�3, 6�: 12

�0, 20�: 20�12, 0�: 0�0, 0�

�24, 8�: 56�36, 0�: 36

−5

−5

40

25

20

−5

50

15

�12, 0�: 12�0, 20�: 0�0, 0�

�40, 0�: 160�24, 8�: 104

−5

−5

40

25

20

−5

50

15

�5, 0�: 10�0, 3�: �3�10, 0�: 20

2 3

1

2

4

x

(0, 3)

(0, 0)

(4, 1)

(5, 0)

y

2 4 6 8

2

4

10

x

(0, 8)

(5, 3)

(10, 0)

y

�4, 1�: 21�0, 0�: 0�5, 3�: 35

2 3

1

2

4

x

(0, 3)

(0, 0)

(4, 1)

(5, 0)

y

2 4 6 8

2

4

10

x

(0, 8)

(5, 3)

(10, 0)

y

�4, 0�: 28�0, 2�: 48�0, 0�: 0�0, 0�: 0

2 6 8

2

4

6

8

x

(0, 8)

(4, 0)

(0, 0)

y

2 3 4 5

−1

1

3

4

x

(0, 2)

(5, 0)

(0, 0)

y

�0, 8�: 64�5, 0�: 30�0, 0�: 0�0, 0�: 0

2 6 8

2

4

6

8

x

(0, 8)

(4, 0)

(0, 0)

y

2 3 4 5

−1

1

3

4

x

(0, 2)

(5, 0)

(0, 0)

y

�0, 800�:�450, 0�:�30, 45�: 2100�60, 20�

�0, 0�: 0�0, 800�:�450, 0�:�60, 20�: 740

�0, 0�:�4, 3�:�4, 0�: 20

�0, 2�:�0, 0�:�4, 3�:�3, 4�:�0, 2�:�0, 0�:�2, 0�:�0, 5�:�0, 0�:�0, 0�:�0, 4�:�5, 0�:�0, 0�:�0, 0�:

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33.

The maximum, 5, occurs at any point on the line segmentconnecting and

34.

The constraints do not form a closed set of points. There-fore, is unbounded.

35.

The constraint is extraneous. Maximum at

36.

The feasible set is empty.

37.

The constraint is extraneous. Maximum at

38.

The maximum, 4, occurs at any point on the line segmentconnecting and

39. 750 units of model A 40. 1000 units of model A1000 units of model B 500 units of model BOptimal profit: $83,750 Optimal profit: $76,000

41. 216 units of $300 model 42. 60 acres of crop A0 units of $250 model 90 acres of crop BOptimal profit: $8640 Optimal profit: $29,550

43. Three bags of brand XSix bags of brand YOptimal cost: $195

44. (a)(b) (c)

(d) gallon of 87 octane and gallon of 93 octane(e) $1.90 per gallon(f) Yes, the cost is lower than the national average of

$1.96 for mid-grade unleaded gasoline.45. 0 tax returns

12 auditsOptimal revenue: $30,000

13

23

y ≥ 0 x ≥ 0

87x � 93y ≥ 89

12

14

14

12

34

34

( (23

13

,

y

x

x � y ≥ 1C � 1.84x � 2.03y

�43, 43�.�0, 2�

x

(0, 2)

(0, 0)

(2, 0)

( (43

43

1

1

,

y

�0, 1�: 42x � y ≤ 4

x

(0, 1)

(1, 0)

(0, 0) 3 4

2

3

y

−2−3 1 2

−2

−1

3

x

y

�0, 7�: 14x ≤ 10

x

(0, 7)

(0, 0)

(7, 0)

2 4 6

2

4

6

10

y

z � x � y

1 2 3 4

3

4

x

(0, 1)

(2, 3)

(0, 0)

y

�2019, 45

19�.�2, 0�

(0, 0)x

1 3

1

2

( (2019

4519

(2, 0)

(0, 3) ,

y

�212 , 0�: 42� 22

3 , 196 �: 271

6

823�22

3 , 196 �:�0, 5�: 25

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(Continued)

46. 42 tax returns5 auditsOptimal revenue: $24,700

47. $62,500 to type A 48. $225,000 to type A$187,500 to type B $225,000 to type BOptimal return: $23,750 Optimal return: $36,000

49. True. The objective function has a maximum value at anypoint on the line segment connecting the two vertices.

50. True. If an objective function has a maximum value atmore than one vertex, then any point on the line segmentconnecting the points will produce the maximum value.

51. (a) (b)

52. (a) (b)

53. 54. 55.

56. 57.

58. 59.

60. 61.

62. 63.

64. 65.

66. 67.

68. ��1, 2, �3��4, 3, �7�e � 9 � �6.282

13 e12�7 � 1.851�ln 6 � �1.792

4 ln 38 � 14.550ln 6 � 1.792ln 4 � 1.386,

ln 3 � 1.099�x � 1��x � 3�

3, x � �1

x2 � 2x � 13

x�x � 2�, x � ±3

1

x � 2, x � 0, �2

9

2�x � 3�, x � 0z � �10x � y

z � 4x � yz � x � yz � x � 5y

t ≥ 6�3 ≤ t ≤ 6

34 ≤ t ≤ 9t ≥ 9

333202CB07_AN.qxd 4/13/06 5:31 PM

Precalculus with Limits, Answers to Review Exercises 17C

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Review Exercises (page 563)1. 2. 3. 4.5. 6. 7.8. 9. 10.

11. 12.13. 14.

15. 3847 units 16. Sales greater than $500,00017. 18.19. 20. 21. 22.23. 24. 25.

26. No solution 27. d, one solution, consistent28. c, infinite solutions, consistent29. b, no solution, inconsistent30. a, one solution, consistent

31. 32.

33. 34. 35.

36. 37.

38. 39.

40. 41.42.43.44.45. (a)

(b) (c) 195.2; yes.

The model is a good fit46. 10 gallons of spray X 47. $16,000 at 7%

5 gallons of spray Y $13,000 at 9%12 gallons of spray Z $11,000 at 11%

48. (a)(b)

49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

63. 64.

65. 66.

67. 68.

69. 70.

−6 −2 4

−2

2

4

8

x

(0, 6)

(−3, 3)

y

x

(2, 3)

(−1, 0)

1−3−4 2 3 4

−2

2

3

4

5

6

y

5 15

5

10

15

x(25, 0)

(25, 25)

(18, 0)

(0, 16)

(0, 25)

(6, 4)

y

x4 12

4

8

12

16

(15, 15)

(6, 3)

(2, 9)

(2, 15)

15, − 32( (

y

4 12 16

4

12

16

x(0, 0)(0, 0)

(8, 0)

(0, 8)

(6, 4)

y

x20 40 80 100

20

40

60

100

(0, 80)

(40, 60)

(60, 0)(0, 0)

y

y

x−4 −3 −2 −1 1 2

4

3

2

−1

−2

−3

−4

3 4x

1 2

4

3

2

1

−2

−1−2−3 3

y

x2−2−4−6

8

6

4

−2

y

x108642−2

6

4

2

8

10

−2

−4

y

2� 1x � 1

�x � 1x2 � 1�

3xx2 � 1

�x

�x2 � 1�2

43�x � 1� �

43�x � 1�2

12�

3x � 1

�x � 3x2 � 1�

12�

3x � 3

�3

x � 3�1 �25

8�x � 5� �9

8�x � 3�

1x � 1

�2

x � 23

x � 2�

4x � 4

Ax

�Bx � Cx2 � 2

�Dx � E

�x2 � 2�2

Ax

�Bx2 �

Cx � 5

Ax � 7

�B

x � 4Ax

�B

x � 20

s � �16t 2 � 20t � 220s � �16t 2 � 150

080

6

130

y � 3x2 � 14.3x � 117.6x2 � y2 � 2x � 2y � 23 � 0x2 � y2 � 4x � 4y � 1 � 0y � 3x2 � 11x � 14

y � 2x2 � x � 5��3a � 2, 5a � 6, a��a � 4, a � 3, a��3

4, 0, �54�

�3a � 4, 2a � 5, a��3817, 40

17, �6317�

�245 , 22

5 , �85��5, �8, 3��2, �4, �5�

�250,000, 95��500,0007

, 1597 �

�85 a �

145 , a��3, 7��0, 0�

�13, �1

2��0.5, 0.8�� 310, 25��5

2, 3�16 feet � 18 feet96 meters � 144 meters

�9.68, �0.84��0, �2�−4

120

4

−6

−6 6

2

��3, 3��0, 0�,��1.41, 10.66��1.41, �0.66�,�1.5, 5��4, �2��2, �1��5, 2�,�0, 0�, �2, 8�, ��2, 8��5, 12��13, 0�,�5, 4�

�115 , 7��0.25, 0.625��3, �3��1, 1�

333202CB07_AN.qxd 4/13/06 5:31 PM

(Continued)

71. 72.

73.

74. (a)

(b)

(c) Answers will vary. Sample answer: and 75. (a)


76. (a)


77. 78.

Minimum at 0 Minimum at 600Maximum at No maximum79. 80.

Minimum at Minimum at 0No maximum Maximum at 60,00081. 82.

Minimum at 0 Minimum at

Maximum at No maximum

83. 72 haircuts, 0 permanents; Optimal revenue: $1800

84. 5 walking shoes 85. Three bags of brand X

2 running shoes Two bags of brand Y

Optimal profit: $138 Optimal cost: $105

86. regular unleaded gasoline

premium unleaded gasoline

Optimal cost: $1.70

87. False. To represent a region covered by an isosceles trape-zoid, the last two inequality signs should be

88. False. An objective function can have no maximum value,one point where a maximum value occurs, or an infinitenumber of points where a maximum value occurs.

≤.

�13�

�23�

�3, 3�: 48

�7, 0�: �14�0, 0�:

y

x

100

75

50

25

1007525

(0, 100)

(25, 50)

(75, 0)

y

x

1

2

1 2 3 4 5 6

3

4

5

6

(0, 4)

(3, 3)

(5, 0)(0, 0)

�500, 500�:�0, 0�:�15, 0�: 26.25

y

x

200

200 400 600 800

400

600(500, 500)

(700, 0)

(0, 750)

(0, 0)

y

x3

3 6 9 12 15 18 21 24 27

6

9

12

15

18

21

24

27 (0, 25)

(5, 15)

(15, 0)

�5, 8�: 47�25, 50�:�0, 0�:

y

x

100

75

50

25

1007525

(0, 100)

(25, 50)

(75, 0)

y

x

3

3 6 9 12 15

6

9

12

15

(0, 10)

(5, 8)

(7, 0)(0, 0)

x

200

150

100

50

100,000 300,000

(200,000, 90)

p = 30 + 0.0003x

p = 130 − 0.0002x


p

x100,000 200,000 300,000

50

75

100

125

150

175

p x= 70 + 0.0002

(300,000, 130)


p

p = 160 − 0.0001x

�16, 9��15, 8�5 10 20 25

5

10

20

25

y

x

�12x10x20x

x

�

�

�

15y ≥20y ≥12y ≥

≥y ≥

300280300

00

y � food Yx � food X,

−400 1600

−400

1600

�20x12x

x

�

�

30y ≤8y ≤

≥y ≥

24,00012,400

00

x−4 2 4 8

−6

−4

2

4

6

32 2

3 3, −( (

32 2

3 3, ( (

y

x2 4 6 8

−2

2

4

6

8

(0, 0) (4, 0)

(6, 4)

y

Precalculus with Limits, Answers to Review Exercises 18

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Precalculus with Limits, Answers to Review Exercises 19

(Continued)

89. 90.

91. 92.

93. 94.

95. 96.

97. An inconsistent system of linear equations has no solution.

98. The lines are distinct and parallel.

99. Answers will vary.

� x � 2y � 32x � 4y � 9

�4x � y � z �

8x � 3y � 2z �

4x � 2y � 3z �

�7

16

31�

2x � 2y � 3z �

x � 2y � z �

�x � 4y � z �

7

4

�1

�x � 2y � z �

2x � y � 4z �

�x � 3y � z �

�7

�25

12�

x � y � z � 6

x � y � z � 0

x � y � z � 2

��x � 4y �

3x � 8y �

10

�21� 3x � y � 7

�6x � 3y � 1

� x � y � 9

3x � y � 11�x � y �

x � y �

2

�14

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333202CB07_AN.qxd 4/13/06 5:31 PM

Chapter Test (page 567)1. 2.3.4. 5.

6.

7. 8.9. 10. No solution

11. 12.

13. 14.

15. 16.

17.

18. Maximum at Minimum at 19. 8%: $20,000 20.

8.5%: $30,00021. 0 units of model I

5300 units of model IIOptimal profit: $212,000

y � �12 x2 � x � 6�0, 0�: 0�12, 0�: 240;

x532

5

3

2

1

−2

−5

1, 15

7, −3 ( (

( (

(1, −3)

−1−2−3−5

y

x−3 6 9 12−6−9−12

−18

3(1, 4)

6

y

(−4, −16)

x−1−2

−2

2

3

31

(0, 0)

(1, 2)

4

1

4

y

�2x

�3x

x2 � 2�

5x

�3

x � 1�

3x � 1

2x2 �

32 � x

�1

x � 1�

3x � 2

�2, �3, 1��2, �1��1, 5�

�0.034, 8.619��1, 12�,

(0.034, 8.619)

(1, 12)

1−1 2 3

4

12

16

y

x

��3, 0�, �2, 5��3, 2�

x−6−9 6 9

−3

−6

3

6

9

12

(−3, 0)

(2, 5)

y

x2 4 6 10

−2

−4

2

4

6

8

(3, 2)

y

�8, 4�, �2, �2��0, �1�, �1, 0�, �2, 1��3, 4�

Precalculus with Limits, Answers to Chapter Test 20

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Precalculus with Limits, Answers to Problem Solving 21

Problem Solving (page 569)1.

Therefore, the triangle is a right triangle.2. 3.4. (a)

Answers will vary.(b)

Answers will vary.

5. (a) One (b) Two (c) Four

6.

Kerry: 57,314,544 votesBush: 60,634,544 votesNader: 354,912 votes

7. 10.1 feet high;

8. Carbon: 12.011 u 9. $12.00

Hydrogen: 1.008 u

10.

5 miles from the airport

11. (a) (b)

12.

13. (a)

(b)

(c) (d) Infinitely many

14.

15.

16. (a) (b)

(c) 142.8 pounds ≤ y ≤ 186.2 pounds

00

45

300

�y ≥ 91 � 3.7xy ≤ 119 � 4.8xx ≥ 0y ≥ 0

t

a

−5

5

30252015105−5

10

20

25

30� a �

0.15a 193a �

t ≤≥

772t ≥

321.911,000

x5 � 1x4 � �5;x3 � 3;x2 � �2;x1 � 2;

��a � 3, a � 3, a�

��11a � 3614

, 13a � 40

14, a�

��5a � 166

, 5a � 16

6, a�

a � 12, b � �4, c � 10

� 2�a � 5

, 1

4a � 1,

1a��3, �4�

�1, 30�

d

t

40

30

20

10

12

12

1 232

−

�d1 � 30td2 � 40�t �

14�

� 252.7 feet long

N � 354,912 B � K � 3,320,000

B � K � N � 118,304,000

�32 a �

72, a�

y

x−2

−4

−2

−1

1

2

3

4

−1 1 52 4 6

�5, 2�

y

x−2

−4

−2

−3

−1

1

3

4

−1 1 32 4 6

y

x−2

−4

−2

−3

−1

1

3

4

−1 1 32 4 5 6

y

x−2

−4

−2

−1

1

2

3

4

−1 1 3 4 5 6

ad � bck1 � �103 , k2 �

163

�8�5�2� �4�5 �2

� 202

a � 8�5, b � 4�5, c � 20

y

x−8 −4

−12

−8

−4

8

12

4 8

(−10, 0)(6, 8)

(10, 0)

a bc

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333202CB07_AN.qxd 4/13/06 5:31 PM

(Continued)

17. (a) (b)

(c) No, because the total cholesterol is greater than 200milligrams per deciliter.

(d) LDL: 140 milligrams per deciliter

HDL: 50 milligrams per deciliter

Total: 190 milligrams per deciliter

(e) answers will vary.�50, 120�; 17050 � 3.4 < 4;

y

x50

50

100

150

200

250

100 150 250

(35,130)

(70, 130)

�xx0

� y

< y

≤ 200≥ 35 ≤ 130

Precalculus with Limits, Answers Problem Solving 22

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precalculus with limits, answers to section 7.1 1

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