5144 demana te ans pp997-1015 - brooklyn high school5144_demana_te_ans_pp997-1015 1/24/06 8:53 am...

ADDITIONAL ANSWERS 997

SECTION 7.1 Exploration 11.

Quick Review 7.1

1. y � �5

3� � �

23

�x 3. x � ��23

�, x � 1 4. x �

Exercises 7.1

17. ��1 �

103�89��, �

3 �

1�0

89�� and ��1 �

103�89��, �

3 �

1�0

89�� 18. ��52 �

675�871��, �

91 �

645�871�� and ��52 �

675�871��, �

91 �

645�871��

45. (a) y � �0.0938x2 � 15.0510x � 28.2375 (b) y � (c) Quadratic: 2006, logistic: 2007

46. (a) y � 2.8763x � 48.4957 (b) y � 3.5148x � 25.0027 47. (a) y � 127.6351x � 2587.0010 (b) y � 31.3732x � 5715.9742

65. (b) 66. (b)

�(�1.29, 2.29) or (1.91, �0.91) �(2.68, 2.68) or (�2.68, �2.68)

SECTION 7.2Exploration 1

1. A � � �; B � � � 2. �A � � �; The order of [0] is 2 � 2. 3. � �8 �111 2

�2 �1�5 �4

�1 22 5

2 15 4

[–9, 9 by –6, 6][–4.7, 4.7 by –3.1, 3.1]

[–5, 30] by [0, 8000][–5, 30] by [0, 8000][–5, 20] by [–10, 100][–5, 20] by [–10, 100]

[0, 30] by [–100, 500][0, 30] by [–100, 500]

353.6473��(1 � 8.6873e�0.1427x)

�5 � �1�0�5��

4

[0, 10] by [–5, 5]

(d) The quadratic regression predicts that the expenditures will eventually be zero.

(e) The logistic regression predicts that the expenditures will eventually level off at about 354 billion dollars.

5144_Demana_TE_Ans_pp997-1015 1/24/06 8:53 AM Page 997

998 CHAPTER 7 Additional Answers

Exploration 21. det A � �a12a21a33 � a13a21a32 � a11a22a33 � a13a22a31 � a11a23a32 � a12a23a31

2. det A � a11a22a33 � a11a23a32 � a12a21a33 � a12a23a31 � a13a21a32 � a13a22a31

3. Since each term in the expansion contains an element from each row and each column, at least one factor in each term is a zero. Therefore, theexpansion will be the sum of n zero terms, or zero.

Exercises 7.2

11. (a) � � (b) � � (c) � � (d) � � 12. (a) � � (b) � �(c) � � (d) � � 13. (a) � � (b) � � (c) � � (d) � �14. (a) � � (b) � � (c) � � (d) � � 15. (a) � � (b) � �(c) � � (d) � � 16. (a) [0 0 �2 3] (b) [�2 �4 2 3] (c) [�3 �6 0 9]

(d) [�5 �10 6 6] 17. (a) � � (b) � � 18. (a) � � (b) � �

19. (a) � � (b) � � 20. (a) � � (b) � �21. (a) � � (b) � � 22. (a) � � (b) � �23. (a) [�8] (b) � � 24. (a) � � (b) [�8]

26. (a) � � (b) The product BA is not defined. 27. (a) � � (b) � �

28. (a) � � (b) � �38. � � 40. � � 44. � �45. (a) The distance from city X to city Y is the same as the distance from city Y to city X. (b) Each entry represents the distance from

city X to city X.

1 11/2 �2

�0.25 0.5 0.250.5 �1.0 0.5

0.25 0.5 �0.25

1 1 �3�0.25 �0.5 1.75

0.25 0.5 �0.75

3 2 �1 �40 1 2 �11 2 �3 32 0 4 �1

�3 2 1 32 1 0 �1

�1 2 3 �44 0 2 �1

1 2 11 0 24 3 �1

�1 3 42 0 11 2 1

1 152 35 �6

�17 15

2 �4 �8�3 6 12

4 �8 �16

�10 5 �158 �4 124 �2 6

�3 �18 �211 2 112 �11 �11

�8 8 50 7 �8

11 4 11

2 1 35 0 0

�18 �3 10

6 �7 �23 7 38 �1 �1

3 �1 �14 164 2 8 �25 3 14 �68 2 0 10

19 �12 10

4 8 �5�5 4 �6�2 �8 6

2 2�11 12

7 �14�8 �10

13 13�2 �16

5 �120 �26

�4 �18�11 �17

�12

�12

�630

�11

�4

�314

16 �13 3 2�14 0 7 10

15 �6 9 3�3 0 6 6

7 �5 2 1�5 0 3 4

3 1 4 13 0 1 0

�18 26 �5

13 5

�9 30 �36 3

�7 12 �25 2

1 1�2 0�1 0

�8 �3 411 2 �8

�8 9 5

�3 0 612 3 �36 0 3

�3 �1 �25 1 �3

�2 3 2

1 1 23 1 16 �3 0

1 154 22

6 9�3 15

1 61 9

3 0�3 1



46. � �; B � 1.1A 48. (a) � � 49. (a) ABT or BAT (b) (A � C)BT 50. (c) C � � �(d) RC 57. [x y]� � 58. [x y]� � 59. [x y]� � 60. [x y]� � 61. [x y]� �62. False. A square matrix A has an inverse if and only if det A � 0. 63. False. It can be negative. For example, the determinant

of A � � � is �1. 70. (c) � �� 0 (d) � �� 0 72. (a) det (xI2 � A) � x2 � (�a22 � a11)x � (a11a22 � a12a21)

(b) They are equal. (c) The coefficient of x is the opposite of the sum of the elements of the main diagonal in A. 73. (b) The constant termequals �det A. (c) The coefficient of x2 is the opposite of the sum of the elements of the main diagonal in A.

SECTION 7.3Exploration 11. x � y � z must equal 60 L. 2. 0.15x � 0.35y � 0.55z must equal 0.40(60) � 24 L.

3. The number of liters of 35% solution must equal twice the number of liters of 55% solution.

4. A � � �; X � � �; B � � � 5. � �6. 3.75 L of 15% acid, 37.5 L of 35% acid, and 18.75 L of 55% acid are required to make 60 L of a 40% acid solution.

Exercises 7.3

1. ��225�, �

72

�, �2� 2. (�3, �3, 2) 3. (1, 2, 1) 4. ��153�, �

1103�, �

7143�� 5. No solution 6. No solution

7. ��92

�, �72

�, 4, ��125�� 8. (2, 1, 0, �1) 9. � � 10. � � 11. � �

12. � �For Exercises 17–20, possible answers are given.

17. � � 18. � � 19. � � 20. � � 23. � �

24. � � 25. � � 26. � � 27. � �28. � �In Exercises 29–32, the variable names are arbitrary.

29. �3x � 2y � �1 30. x � z � 2w � �3 31. 2x � z � 3 32. 2x � y �2z � 4�4x � 5y � 2 2x � y � w � 4 �x � y � 2 �3x �2z � �1

�x � y � 2z � 0 2y � 3z � �1

57

�25

3�1

�345

�11

�1

1�2�3

�502

210

14

�1�2

11

�40

31

1�3

2

1�4�1

�310

2�1

3

�100

100

�200

100

3�1

�12

01

10

�21

3�1

21

10

�4�0.6�9.2

301

210

100

�310

200

100

�1�1.2

1

310

100

41

�2

�6�4

1

23

�3

10�3�2

�1021

01

�3

2�3�2

�321

11

�3

4�3

4

�62

�8

210

3.7537.5

18.75

60240

xyz

10.55�2

10.35

1

10.15

0

y3

y1

y2

x3

x1

x2

111

y3

y1

y2

x3

x1

x2

111

1 02 �1

01

c0

0a

10

�10

0�1

10

01

01

�10

$1600$900$500$100

$1000

$15,550 $21,919.54$8070 $11,439.74$8740 $12,279.76

132 77165 12188 176


33. (2, �1, 4); � � 34. (�1, 2, �3); � � 35. (�2, 3, 1) 36. (7, 6, 3)

37. No solution 38. (z � 2, �z � 1, z) 39. (2 � z, 1 � z, z) 40. (z � 53, z � 26, z) 43. (z � w � 2, 2z � w � 1, z, w)

44. (z � w, w � 3, z, w) 45. � � � � � � � 46. � � � �� 51. (�2, �5, �7) 52. (3, �0.5, 0.5) 59. �2 � �

32

�z, ��12

�z � 4, z� 60. ��15

�z � 1, �35

�z � 2, z� 61. (�2w � 1, w � 1, �w, w)

62. (w, �2w � 1, �w � 1, w) 63. (�w � 2, �z � 0.5, z, w) 64. (z � 3w � 1, �2z � 2w � 4, z, w) 65. No solution66. (1, 1� w, 6w � 2, w) 68. f(x) � 3x3 � x2 � 2x � 5 70. f(x) � (�c � 4)x3 � x2 � cx � 1, for any c71. (a) y � 2.0734x � 234.0268 (b) y � 3.5302x � 141.7246 72. (a) y � 5.1670x � 166.2935 (b) y � 6.3140x � 78.3593 (c) 2046

93. (b) 94. (b)

SECTION 7.4 Quick Review 7.4

1. �x2 �

3x4�

x �

53

� 2. �x2

3�

(x5�

x �

1)4

� 3. �4xx3

2

�

�

26xx2 �

�

x1

� 4. �3x

(

2

x2

�

�

x1�

)2

2� 5. 3x2 � 2 � �

x �

32

� 6. 2x � 1 � �x2

3�

x�

x �

26

�

7. (x � 1)(x � 3)(x2 � 4) 8. (x � 1)2(x � 2)(x � 5)

Exercises 7.4

1. �Ax

1� � �x

A�

2

2� � �

xA�

3

2� 2. �

x

B2

1

�

x �

x �

C1

1� � �

(x2B

�2x

x�

�

C12

)2� � �xB2

3

�

x �

x �

C3

1� 3. �

Ax

1� � �Ax2

2� � �Ax3

3� � �x

A�

4

1� � �

(x �

A5

1)2� � �Bx12

x�

� C9

1�

4. �x

A�

1

1� � �

(x �

A2

1)2� � �(x �

A3

1)3� � �xB2

1

�

x �

x �

C1

1� � �

(x2

B�2x

x�

�

C12

)2� � �(x2

B�3x

x�

�

C13

)3� 7. �x2 �

31

� � �(x22

x�

�

11)2�

9. �x �

12

� � �(x �

22)2� � �

(x �

12)3� 10. �

�

x32

x�

�

47

� � �8xx2

�

�

197

� 11. �x �

23

� � �(x

�

�

13)2� � �

3xx2 �

�

21

� � �(x

x2

�

�

22)2�

12. �x�

�

21

� � �(x �

31)2� � �

x �

14

� � �(x �

34)2� 13. �

x �

15

� � �x�

�

13

� 14. �x �

13

� � �x�

�

17

� 15. �x �

21

� � �x�

�

21

�

16. �x �

13

� � �x�

�

13

� 19. �x �

13

� � �x�

�

24

� 20. �x �

45

� � �x �

32

� 21. �x

�

�

23

� � �2x

5� 1� 22. �

x �

31

� � �2x

2� 3�

23. �x2 �

21

� � �(x2 �

31)2� 24. �

x2 �

31

� � �(x2 �

11)2� 25. �

2x

� � �x�

�

11

� � �(x �

21)2� 26. �

(x7�

/33)2� � �

�

x2�

5/39

� � �25

x/9�

27. �1x

� � ��

x21� � �

x �

23

� 28. �2x

� � ��

x21� � �

x �

34

� 29. �x2

2�

x2

� � �(x2

�

�

12)2� 30. �

x2

3�

x2

� � �(x2

�

�

12)2�

31. �x �

21

� � �x2 �

�

xx� 1

� 32. �x �

31

� � �x2 �

�

xx� 1

�

[–2.7, 6.7] by [–5.1, 1.1][–1, 8.4] by [–3.1, 3.1]

[–3, 35] by [0, 400][–3, 35] by [0, 400][–3, 30] by [0, 400][–3, 30] by [0, 400]

23

�3

xyz

1�1

1

�7�3

1

521

�31

xy

5�2

21

128

�3

�3�2

1

210

100

8�5

4

1�1

1

�210

100

1000 ADDITIONAL ANSWERS



33. 2 � �xx2

�

�

51

�; �x �

31

� � �x�

�

21

�

Graph of (2x2 � x � 3)/(x2 � 1): Graph of 3/(x � 1): Graph of �2/(x � 1):

34. 3 � �2xx2�

�

142

�; �x �

42

� � �x�

�

22

�

Graph of (3x2 � 2x)/(x2 � 4) Graph of 4/(x � 2): Graph of �2/(x � 2):

35. x � 1 � �xx2�

�

2x

�; �x �

31

� � ��

x2�

Graph of y � (x3 � 2)/(x2 � x) Graph of y � x � 1: Graph of y � 3/(x � 1): Graph of y � �2/x:

36. x � 1 � �xx2

�

�

2x

�; �x �

31

� � ��

x2�

Graph of y � (x3 � 2)/(x2 � x): Graph of y � x � 1: Graph of y � 3/(x � 1): Graph of y � �2/x:

47. True. The behavior of f near x � 3 is the same as the behavior of y � �x �

13

� and limx → 3�

�x �

13

� � ��.

48. True. The behavior of f for x large is the same as the behavior of y � �1 and limx → ∞

(�1) � �1.

[–4.7, 4.7] by [–15, 10][–4.7, 4.7] by [–15, 10][–4.7, 4.7] by [–15, 10][–4.7, 4.7] by [–15, 10]

[–4.7, 4.7] by [–10, 15][–4.7, 4.7] by [–10, 15][–4.7, 4.7] by [–10, 15][–4.7, 4.7] by [–10, 15]

[–4.7, 4.7] by [–10, 10][–4.7, 4.7] by [–10, 10][–4.7, 4.7] by [–10, 10]

[–4.7, 4.7] by [–10, 10][–4.7, 4.7] by [–10, 10][–4.7, 4.7] by [–10, 10]


SECTION 7.5Quick Review 7.51. 2. (6, 0); (0, 3) 3. (20, 0); (0, 50) 4. (30, 0); (0, �20)

5. (30, 60) 6. (70, 20) 7. (10, 140) 8. (2, 4) 9. (3, 3)

Exercises 7.51. Graph (c); boundary included 2. Graph (f); boundary excluded 3. Graph (b); boundary included4. Graph (d); boundary excluded 5. Graph (e); boundary included 6. Graph (a); boundary excluded7. 8. 9.

boundary x � 4 included boundary y � �3 included boundary 2x � 5y � 7 included10. 11. 12.

boundary 3x � y � 4 excluded boundary y � x2 � 1 excluded boundary y � x2 � 3 included

y

5

x5

y

9

x5

y

5

x5

y

5

x5

y

5

x5

y

5

x5

x

y

50

50

x

y

50

50

x

y

10

5

x

y

5

5




13. 14. 15.

boundary x2 � y2 � 9 excluded boundary x2 � y2 � 4 included boundary y � (ex � e�x)/2 included16. 17. 18.

boundary y � sin x excluded19. 20. 21. 22.

23. 24. 25. 26.

27. x2 � y2 4 28. x2 � y2 4 29. y ��12

�x � 5 30. y ��52

�x � 6

y �x2 � 1 y 0y ��

32

�x � 9 y ��13

�x � �35

�

x 0 y 0y 0 y 0

31. The minimum is 0 at (0, 0); the maximum is 880/3 at (160/3, 80/3). 32. The minimum is 0 at (0, 0); the maximum is 967.5 at (22.5, 67.5). 33. The minimum is 162 at (6, 30); there is no maximum.34. The minimum 315 at (16, 3); the maximum is 505 at (32, 1). 35. The minimum is 24 at (0, 12); there is no maximum.

y

80

x80

y

9

x9

y

90

x90

y

90

x90

y

5

x5

y

5

x5

y

4

x3

y

16

x10

y

10

x10

y

10

x10

y

2

x2π

y

8

x5

y

5

x5

y

5

x5


36. The minimum is 34 at (8, 2); there is no maximum. 37. �13.48 tons of ore R and �20.87 tons of ore S; $1926.2038. 1.6 units of food substance A and 9.6 units of food substance B; $10.88 39. x operations at Refinery 1 and y operations at Refinery 2 suchthat 2x � 4y � 320 with 40 x 120. 40. 2000 units of product A and 1000 units of product B 41. False. It is a half-plane.42. True. The line 3y � 2x � 5 borders the half-plane determined by 2x � 3y � 5.

49. y1 � 21 � �x9

2��; y2 � �21 � �

x9

2�� 50. y1 � �x2� �� 4�; y2 � ��x2� �� 4�

51. 52.

CHAPTER 7 REVIEW EXERCISES1. (a) � � (b) � � (c) � � (d) � � 2. (a) � � (b) � �(c) � � (d) � � 3. � �; not possible

4. Not possible; � � 6. Not possible; � � 7. � �; � �

8. � �; � � 9. AB � BA � I4 10. AB � BA � I3 11. � �12. � � 15. � � 16. � � 17. � � 18. � �23. (�z � w � 2, w � 1, z, w) 24. (�w � 2, �z � w, z, w) 25. No solution 26. ((1/4)z � 3/4, (7/4)z � 5/4, z)27. (�2z � w � 1, z � w � 2, z, w) 29. (9/4, �3/4, �7/4) 30. (1/2, �5/2, �5/2) 39. (x, y) � (0.14, �2.29)40. (x, y) � (�1, �2.5) or (x, y) � (3, 1.5) 41. (x, y) � (�2, 1) or (x, y) � (2, 1)42. (x, y) � (�1.47, 1.35) or (x, y) � (1.47, 1.35) or (x, y) � (0.76, �1.85) or (x, y) � (�0.76, �1.85) 43. (x, y) � (2.27, 1.53)44. (x, y) � (4.62, 2.22) or (x, y) � (1.56, 1.14) 45. (a, b, c, d ) � (17/840, �33/280, �571/420, 386/35)

46. (a, b, c, d, e) � (19/108, �29/18, 59/36, 505/54, �68/9) 47. �x �

11

� � �x �

24

� 48. �x �

62

� � �x�

�

51

�

49. �x�

�

12

� � �x �

11

� � �(x �

21)2� 50. �

x �

21

� � �x �

12

� � �(x

�

�

32)2� 51. �

x �

21

� � �3x2

x�

�

14

� 52. �x �

12

� � ��

x22x�

�

41

�

2�1

1

001

010

100

8�11

5

001

010

100

12

�3

001

010

100

2�1

0

010

100

0.20.60.2

0.2�0.4

0.2

�0.4�0.2

0.6

�104

�1

61

�278

�5�124

�7

�20

10�3

0221

11

�10

�23

�13

101

�2

110

�1

2012

01

�21

3�2

3�1

4�3�1

21

�2

�321

�34

�1

2�3

1

12

�2

�1342

�3�1�3�1

�43

�13

151

�5

1124

�7�12

�30

�2�15�3

�3�12

4

514

�17

8�1

4

�46

�2

24

�4

�6�8

6

�4�2

0

�2�6�2

�1�5

1

15

�7

3�1

2

604

�113

531

13

�2

11�6

�74

�60

2�8

4�3

�30

23

18

y

5

–5

–5x

5

y

5

x5




57. 58. 59. 60.

Corners at (0, 90), (90, 0), Corners at (0, 3), (0, 7),(360/13, 360/13). (30/13, 70/13), (3, 0), (5, 0). Boundaries included. Boundaries included.

61. 62. 63. 64.

Corners at �(0.92, 2.31) Corners at (�2.41, 3.20) Corners at �(�1.25, 1.56) No corner points. and �(5.41, 3.80). and (2.91, 0.55). and �(1.25, 1.56). Boundaries included.Boundaries excluded. Boundaries included. Boundaries included.

65. The minimum is 106 at (10, 6); there is no maximum. 66. The minimum is 138 at (8, 10); there is no maximum.67. The minimum is 205 at (10, 25); the maximum is 292 at (4, 40).68. The minimum is 600 at (20, 30); the maximum is 1680 at (0, 120). 69. (a) �(2.12, 0.71) (b) �(�0.71, 2.12)

70. (a) y � 11.4428x � 116.681 (b) y �

71. (a) y � 12.2614x � 979.5909 (b) y � 19.8270x � 893.9566

[–5, 30] by [0, 2000][–5, 30] by [0, 2000]

[–5, 20] by [0, 400][–5, 20] by [0, 400]

294.846��(1 � 1.6278e�0.17864)

y

7

x5

y

5

x5

y

10

x5

y

7

x8

y

9

x9

y

90

x90

y

5

x5

y

5

x5

(c) The two regressions will be equal when the graph of their dif-ference is 0. This occurs approximately at x � 3 (1993) andx � 10.2 (early 2000).

(d) Both fit the data fairly well. For beyond 2000 the logisticregression is likely better.


M F �18 65

72. (a) � � (b) � � (c) � � (d) � � This matrix provides total numbers of males and females who are under 18 or are 65 or older in all three states.

(e) 6.7 million; 3.4 million

Chapter 7 Project1. 4.

Males: y � 1.7585x � 119.5765

Females: y � 1.6173x � 126.4138

SECTION 8.1Exploration 11. The axis of the parabola with focus (0, 1) and directrix y � �1 is the y-axis because it is perpendicular to y � �1 and passes through (0, 1).The vertex lies on this axis midway between the directrix and the focus, so the vertex is the point (0, 0).3. {(�2�6�, 6), (�2�5�, 5), (�4, 4), (�2�3�, 3), (�2�2�, 2), (�2, 1), (0, 0), (2, 1), (2�2�, 2), (2�3�, 3), (4, 4), (2�5�, 5), (2�6�, 6)}

Exploration 21. 2. 3.

The equation of the axis is x � 2.

4. The focal length p is �3, and the 5. 6.focal width �4p� is 12.

y

x

10

10F(2, –2) B(8, –2)

A(–4, –2)

V(2, 1)

y = 4

x = 2

y

x

10

10F(2, –2)

B(8, –2)A(–4, –2)

V(2, 1)

y = 4

x = 2

y

x

10

10F(2, –2)

y = 4

x = 2

V(2, 1)

y

x

10

10F(2, –2)

y = 4

x = 2

y

x

10

10F(2, –2)

y = 4

[–10, 120] by [30, 150][–5, 15] by [120, 160]

6.93.4

6.73.4

0.1060.1700.140

0.2650.2310.228

10.617.014.0

26.523.122.8

CAFLRI

18.08.6

0.56

17.58.4

0.54

CAFLRI


5144_Demana_TE_Ans_pp997-1015 1/24/06 8:53 AM 06

Quick Review 8.1

2. �(a � 2�)2 � (b� � 3)2� 5. y � 6 � �(x � 1)2 6. y � �129� � 2�x � �

32

��2

7. Vertex: (1, 5); f(x) can be obtained from g(x) by stretching 8. Vertex: (3, 19); f(x) � �2(x � 3)2 � 19. f(x) can bex2 by 3, shifting up 5 units, and shifting right 1 unit. obtained from g(x) by stretching x2 by 2, reflecting over

the x-axis, shifting up 19 units and shifting right 3 units.

Exercises 8.1

1. Vertex: (0, 0); Focus: �0, �32

��; Directrix: y � ��32

�; Focal width: 6 2. Vertex: (0, 0); Focus: (�2, 0); Directrix: x � 2;

Focal width: 8 3. Vertex: (�3, 2); Focus: (�2, 2); Directrix: x � �4; Focal width: 4 4. Vertex: (�4, �1);

Focus: ��4, ��52

��; Directrix: y � �12

�; Focal width: 6 5. Vertex: (0, 0); Focus: �0, ��13

��; Directrix: y � �13

�; Focal width: �43

�

6. Vertex: (0, 0); Focus: ��45

�, 0�; Directrix: x � ��45

�; Focal width: �156� 27. (x � 2)2 � 16(y � 1) 28. (x � 3)2 � �20(y � 3)

29. (y � 4)2 � �10(x � 1) 30. (y � 3)2 � 5(x � 2)

31. 32. 33.

34. 35. 36.

37. 38. 39. 40.

[–2, 8] by [–3, 3][–8, 2] by [–2, 2][–10, 10] by [–8, 2][–4, 4] by [–2, 18]

y

10

x20

y

10

x6

y

10

x2

y

5

x5

y

5

x5

y

5

x5

[–2, 7] by [–10, 20][–3, 4] by [–2, 20]



41. 42. 43. 44.

45. 46. 47. 48.

49. Completing the square, the equation becomes (x � 1)2 � y � 2, a parabola with vertex (�1, 2), focus (�1, 9 4), and directrix

y � 7 4. 50. Completing the square, the equation becomes (x � 1)2 � 2(y � 7 6), a parabola with vertex (1, 7 6), focus (1, 5 3),

and directrix y � 2 3. 51. Completing the square, the equation becomes (y � 2)2 � 8(x � 2), a parabola with vertex (2, 2),

focus (4, 2), and directrix x � 0. 52. Completing the square, the equation becomes (y � 1)2 � �4(x � 13 4), a parabola with

vertex (13 4, 1), focus (9 4, 1), and directrix x � 17 4. 53. (y � 2)2 � �6x 54. (y � 3)2 � 2(x � 1)

55. (x � 2)2 � �4(y � 1) 56. (x � 1)2 � 8(y � 3) 57. The derivation only requires that p is a fixed real number.

58. Let P(x, y) be a point on the parabola with focus (p, 0) and directrix x � �p.

Distance from P(x, y) to (p, 0) � distance from P(x, y) to x � �p

�(x � p�)2 � (y� � 0)2� � �(x �(��p))2 �� (y � y�)2�(x � p)2 � (y � 0)2 � (x �(�p))2 � (y � y)2

x2 � 2px � p2 � y2 � x2 � 2px � p2

y2 � 4px

59. The filament should be placed 1.125 cm from the vertex along the axis of the mirror. 60. The receiving antenna should be placed0.78125 ft, or 9.375 inches, from the vertex along the axis of the reflector. 61. The electronic receiver is located 2.5 units from the vertexalong the axis of the parabolic microphone. 62. The light bulb should be placed 3 units from the vertex along the axis of the headlight.

63. Starting at the leftmost tower, the lengths of the cables are: � {79.44, 54.44, 35, 21.11, 12.78, 10, 12.78, 21.11, 35, 54.44, 79.44}.

65. False. Every point on a parabola is the same distance from its focus and its directrix. 66. False. The directrix of a parabola is perpendicular to the parabola’s axis.

71. (a)–(c) 72. (a)–(d) y

5

x5

(0, 1)P(x, n – 1)

y = n – 1

y = –1

x2 + (y – 1)2 = n2

y

x

APF

l

[–20, 28] by [–10, 22][–13, 11] by [–10, 6][–17, 7] by [–7, 9][–22, 26] by [–19, 13]

[–15, 5] by [–15, 5][–2, 6] by [–40, 5][–12, 8] by [–2, 13][–10, 15] by [–3, 7]



73. (a) (b) (c)

(d)


1. x � �2 � 3 cos t and y � 5 � 7 sin t; cos t � �x �

32

� and sin t � �y �

75

�; cos2 t � sin2 t � 1 yields the equation

�(x �

92)2

� � �(y �

495)2

� � 1.

2. 3. Example 1: x � 3 cos t and y � 2 sin tExample 2: x � 2 cos t and y � �13� sin tExample 3: x � 3 � 5 cos t and y � �1 � 4 sin t

4.

[–3, 9] by [–6, 4][–6, 6] by [–4, 4][–4, 4] by [–3, 3]

[–17.5, 12.5] by [–5, 15]

Plane Line

Axis

Generator

PlaneCircle Single line

Two parallel lines

Axis Generator

Cylinder



5. Example 1: x � 3 cos t, y � 2 sin t; cos t � �3x

�, sin t � �2y

�; cos2 t � sin2 t � 1 yields �x9

2

� � �y4

2

� � 1, or 4x2 � 9y2 � 36.

Example 2: x � 2 cos t, y � �13� sin t; cos t � �2x

�, sin t � ��

y

13��; sin2 t � cos2 t � 1 yields �

1y3

2

� � �x4

2

� � 1.

Example 3: x � 3 � 5 cos t, y � �1 � 4 sin t; cos t � �x �

53

�, sin t � �y �

41

�; cos2 t � sin2 t � 1 yields �(x �

253)2

� � �(y �

161)2

� � 1.

Exploration 22. a � 9 cm, b � 8.94 cm, c � 1 cm, e � 0.11, b a � 0.993. a � 8 cm, b � 7.75 cm, c � 2 cm, e � 0.25, b a � 0.97; a � 7 cm, b � 6.32 cm, c � 3 cm, e � 0.43, b a � 0.90; a � 6 cm,b � 4.47 cm, c � 4 cm, e � 0.67, b a � 0.754. The ratio b a decreases slowly as e � c a increases. The ratio b a is the height-to-width ratio, which measures the shape of the ellipse. Theeccentricity ratio e � c a measures how off-center the foci are.5.

b a � �1 � e2�

Quick Review 8.2

3. y � ��32

��4 � x2� 4. y � ��56

��36 � x�2� 9. x � �3 ��

215�

� 10. x � �1 � �72

��

Exercises 8.21. Vertices: (4, 0), (�4, 0); Foci: (3, 0), (�3, 0) 2. Vertices: (0, 5), (0, �5); Foci: (0, 2), (0, �2)

3. Vertices: (0, 6), (0, �6); Foci: (0, 3), (0, �3) 4. Vertices: (�11�, 0), (��11�, 0); Foci: (2, 0), (�2, 0)

5. Vertices: (2, 0), (�2, 0); Foci: (1, 0), (�1, 0) 6. Vertices: (0, 3), (0, �3); Foci: (0, �5�), (0, ��5�)

9. (a) 10. (b)11. 12. 13. 14.

15. 16. 17. 18.

[–11.75, 11.75] by [–8.1, 8.1][–9.4, 9.4] by [–6.2, 6.2]

y

2

x6

y

8

x4

y

10

x10

y

5

x5

y

10

x10

y

10

x10

[–0.3, 1.5] by [0, 1.2]



19. 20. 21. �y9

2

� � �x4

2

� � 1 22. �4x9

2� � �

2y5

2� � 1 25. �

2y5

2� � �

1x6

2� � 1

26. �4x9

2� � �

1y6

2� � 1 27. �

3y6

2� � �

1x6

2� � 1 28. �

2x5

2� � �

y4

2� � 1

29. �2x5

2� � �

1y6

2� � 1 30. �

1y6

2

9� � �

1x4

2

4� � 1

31. �(y �

362)2

� � �(x �

161)2

� � 1 37. Center: (�1, 2); Vertices: (�6, 2), (4, 2); Foci: (�4, 2), (2, 2)

38. Center: (3, 5); Vertices: (3 � �11�, 5); Foci: (1, 5), (5, 5) 39. Center: (7, �3); Vertices: (7, 6), (7, �12); Foci: (7, �3 � �17�)

40. Center: (�2, 1); Vertices: (�2, 6), (�2, �4); Foci: (�2, 4), (�2, �2)

41. 42. 43. 44.

x � 2 cos t, y � 5 sin t x � �30� cos t, x � 2�3� cos t � 3, x � �6� cos t � 2,

y � 2�5� sin t y � �5� sin t � 6 y � �15� sin t � 1

45. Vertices: (1, �4), (1, 2); Foci: (1, �1 � �5�); Eccentricity: ��35�� 46. Vertices: (2 � �5�, �3); Foci: (2 � �2�, �3);

Eccentricity: �25

�� 47. Vertices: (�7, 1), (1, 1); Foci: (�3 � �7�, 1); Eccentricity: ��

47�� 48. Vertices: (4, �10), (4, �6);

Foci: (4, �8 � �3�); Eccentricity: ��

23�� 49. �

(x �

162)2

� � �(y �

93)2

� � 1 50. �(x �

164)2

� � �(y �

92)2

� � 1

53. a � 237,086.5, b � 236,571, c � 15,623.5, e � 0.066 56. Venus: �11,7

x0

2

7.24� � �

11,7y0

2

6.70� � 1; Mars: �

51x,9

2

38� � �

51y,4

2

85� � 1

60. Place the source and the patient at opposite foci, 12 inches from the center along the major axis.

63. (a) Approximate solutions: (�1.04, �0.86), (�1.37, 0.73)

(b) �� , ��1 �

1�6

161��, �� , �

�1 �

16�161��

64. The word can mean a shape that is generally round but not exactly circular, or a person who does not follow conventional norms of behavior.65. False. The distance is a(1 � e). 66. True, because a2 � b2 � c2 in any ellipse.

71. (a) When a � b � r, A � �ab � �rr � �r2 and P � �(2r) � (3 � �(4r)(4r�)�/(2r)) � �(2r) � (3 � 2) � 2�r.

73. (a)

(b) y2/4 � (x � 3)2 � 1

74. (a) Because cos2(�t � (�/2)) � sin2 (�t � (�/2)) �1, the graph of y2/(9�2) � (x � 5)2/9 � 1 is an ellipse. (b) The pendulum begins its swing

(at t � 0 sec) at its maximum distance of 8 ft from the motion detector; it takes 1 sec for the pendulum to swing from this maximum distance to

the minimum distance of 2 ft and another second to swing back to the 8-ft position. It is moving fastest when it is at the halfway position of 5 ft

from the detector. 76. The graph consists of the single point (h, k). It is like an ellipse with semimajor and semiminor axes both equal to 0.

See Figure 8.2.

[–4.7, 4.7] by [–3.1, 3.1]

�94 � 2��161��

8�94 � 2��161��

8

[–3, 7] by [–5, 3][–8, 2] by [0, 10][–6, 10] by [–6, 5][–8, 8] by [–6, 6]

[–9, 17] by [–6, 6][–17, 4.7] by [–3.1, 3.1]




1. x � �1 � 3 cos t � �1 � 3 sec t and y � 1 � 2 tan t; sec t � �x�

31

� and tan t � �y �

21

�; sec2 t � tan2 t � 1 yields the equation

�(x �

91)2

� � �(y �

41)2

� � 1.

2. In Connected graphing mode, pseudo-asymptotes appear because thegrapher connects computed points by line segments regardless of whether this makes sense.

3. Example 1: x � 3 cos t, y � 2 tan t; Example 2: x � 2 tan t, y � �5� cos t; Example 3: x � 3 � 5 cos t, y � �1 � 4 tan t; Example 4: x ��2 � 3 cos t, y � 5 � 7 tan t

4.

5. Example 1: x � 3 cos t � 3 sec t, y � 2 tan t; sec t � x 3, tan t � y 2; sec2 t � tan2 t � 1 yields x2 9 � y2 4 � 1,

or 4x2 � 9y2 � 36. Example 2: x � 2 tan t, y � �5� cos t � �5� sec t; tan t � x 2, sec t � y �5�; sec2 t � tan2 t � 1

yields y2 5 � x2 4 � 1. Example 3: x � 3 � 5 cos t � 3 � 5 sec t, y � �1 � 4 tan t; sec t � (x � 3) 5, tan t � (y � 1) 4;

sec2 t � tan2 t � 1 yields (x � 3)2 25 � (y � 1)2 16 � 1. Example 4: x � �2 � 3 cos t � �2 � 3 sec t, y � 5 � 7 tan t;

sec t � (x � 2) 3, tan t � (y � 5) 7; sec2 t � tan2 t � 1 yields (x � 2)2 9 � (y � 5)2 49 � 1.

Quick Review 8.33. y � ��

43

��9 � x2� 4. y � ��13

��x2 � 3�6�

Exercises 8.31. Vertices: (�4, 0); Foci: (��23�, 0) 2. Vertices: (0, �5); Foci: (0, ��46�) 3. Vertices: (0, �6); Foci: (0, �7)

4. Vertices: (�3, 0); Foci: (�5, 0) 5. Vertices: (�2, 0); Foci: (��7�, 0) 6. Vertices: (�2, 0); Foci: (��13�, 0)

11. 12. 13. y

x

15

20

y

25

x20

y

15

x20

[–9.4, 9.4] by [–6.2, 6.2] [–9.4, 9.4] by [–6.2, 6.2] [–7.4, 11.4] by [–6.2, 6.2] [–20, 18] by [–8, 18]

[–9.4, 9.4] by [–6.2, 6.2][–9.4, 9.4] by [–6.2, 6.2]



14. 15. 16.

17. 18. 19. 20.

21. 22. 26. �9x/

2

4� � �

9y1

2

/4� � 1 27. �

2x5

2� � �

7y5

2� � 1

28. �1y6

2

� � �2x0

2

� � 1 29. �1y4

2

4� � �

2x5

2

� � 1

30. �x9

2� � �

2y7

2� � 1 31. �

(y �

41)2

� � �(x �

92)2

� � 1

32. �(x �

361)2

� � �(y �

253)2

� � 1 33. �(x �

92)2

� � �(y �

163)2

� � 1 34. �(y �

815/4/2)2

� � �(7x2�

9/624)2

� � 1 35. �(x �

41)2

� � �(y �

52)2

� � 1

36. �(y �

49/54.5)2

� � �(x �

183)2

� � 1 37. �(y �

256)2

� � �(x �

753)2

� � 1 38. �(x �

91)2

� � �(y �

274)2

� � 1

39. Center: (�1, 2); Vertices: (11, 2), (�13, 2); Foci: (12, 2), (�14, 2) 40. Center: (�4, �6); Vertices: (�4 � 2�3�, �6);

Foci: (1, �6), (�9, �6) 41. Center: (2, �3); Vertices: (2, 5), (2, �11); Foci: (2, �3 � �145�) 42. Center: (�5, 1);

Vertices: (�5, �4), (�5, 6); Foci: (�5, �5), (�5, 7)

43. 44. 45. 46.

47. 48.

Vertices: (3, �2), (3, 4); Foci: (3, 1 � �13�); e � ��

313�� Vertices: ��

32

�, �4�, ��52

�, �4�; Foci: �2 � ��

613��, �4�; e � �

�313��

[–9.4, 9.4] by [–5.2, 7.2][–9.4, 9.4] by [–5.2, 7.2]

[–7.4, 11.4] by [–7.2, 5.2][–12.4, 6.4] by [–0.2, 12.2][–14.1, 14.1] by [–9.3, 9.3][–14.1, 14.1] by [–9.3, 9.3]

[–11.4, 7.4] by [–3.2, 9.2][–9.4, 9.4] by [–3.2, 9.2]

[–9.4, 9.4] by [–6.2, 6.2][–9.4, 9.4] by [–6.2, 6.2][–18.8, 18.8] by [–12.4, 12.4][–9.4, 9.4] by [–6.2, 6.2]

y

3

x6

y

4

x3

y

x

15

20



49. 50.

Vertices: (0, 1), (4, 1); Foci: (2 � �13�, 1); e � ��

213�� Vertices: (�3, �2), (�3, 4); Foci: (�3,

1 � �34�); e � ��

334��

51. �x4

2

� � �51y6

2

� � 1 52. �y2

2

� � �x4

2

� � 1

54. (a) (b) 55. a � 1440, b � 600, c � 1560, e � 13 12; The Sun is centered at (1560, 0).

56. a � 156.8, b � 252, c � 296.8, e � 53 28; The Sun is centered at (296.8, 0).

57. A bearing and distance of about 40.29° and 1371.11 miles, respectively

58. A bearing and distance of about 50.11° and 15,628.2 ft (2.89 mi), respectively

59. (�2, 0), (4, 3�3�) 60. Four solutions: (�2�2�, �1) 61. (a)

(b) ��10�62491

��, �10�62411

��62. One possibility: Escape speed is the minimum speed one object needs to achieve in order to break away from the gravity of

another object. 63. True, because c – a � ae � a. 64. True. For an ellipse, b2 � c2 � a2.

69. (a–d) 70. �16

x0

2

0� � �

20y0

2

0� � 1

72. The graph consists of two intersecting slanted lines through (h, k). Its symmetry is like that of a hyperbola. Figure 8.2 shows the relationship between an ordinary hyperbola and two intersecting lines.

y

x5

[–9.4, 9.4] by [–6.2, 6.2][–9.4, 9.4] by [–4.2, 8.2]

Four solutions:(�2.13, �1.81)

[–9.4, 9.4] by [–6.2, 6.2]

y

4

x4

y

4

x4

[–12.4, 6.4] by [–5.2, 7.2][–9.4, 9.4] by [–6.2, 6.2]



SECTION 8.4Exercises 8.41. y � �5 � ��x2 �� 6x ��7� 2. y � 1 � 2��x2 �� 6x ��5� 3. y � 4 � 2�2x � 2� 4. y � �5 � 0.5�x2 � 6�x � 19�1�

5. y � 4 x 6. y � �3 x 7. y � 8 (x � 1) 8. y � 2x2 (5x � 1)

9. y � �16

�(x � 4 � ��23x2� � 28x� � 88�) 10. y � �18

�(10 � 3x � �25x2 �� 20x �� 420�)

11. y � �14

�(x � 1 � �3(�x2�� 6x �� 9)�) 12. y � x � �32

� � �12

� �39 � 2�x� 13. x2 � �4y 14. y2 � 8x 15. �x9

2� � �

1

y

6

2� � 1

16. �1x6

2� � �

y9

2� � 1

21. Hyperbola: �(y �

91)2

� � �(x �

41)2

� � 1; �(y

9�)2� � �

(x4�)2� � 1 22. Ellipse: �

(x �

33)2

� � �(y �

24)2

� � 1; �(x

3

�)2

� � �(y

2

�)2

� � 1

y�

3

x�4

y�

8

x�5

[–10, 30] by [–5, 20][–2, 8] by [–3, 3]

[–10, 10] by [–8, 8][–4.7, 4.7] by [–3.1, 3.1]

[–1, 1.4] by [–0.4, 0.8][–10, 12] by [–12, 12][–9.4, 9.4] by [–6.2, 6.2][–9.4, 9.4] by [–6.2, 6.2]

[–37.6, 37.6] by [–24.8, 24.8][–19.8, 17.8] by [–8.4, 16.4][–9.4, 9.4] by [–6.2, 6.2][–6.4, 12.4] by [–11.2, 1.2]


20. (x�, y�) � (�5 � �2�, �1)


5144 demana te ans pp997-1015 - brooklyn high school5144_demana_te_ans_pp997-1015 1/24/06 8:53 am...

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