exercise 1-8 - mcgraw hill education€¦ · 5 12 01 1 2 1 5 5 2 2 2 1 2 1 5 5 2 2 2 x 4 0 2 [x...

Answers A-81

29. x � � � i 31. y � � 33. t � � , � 35. z � � 37. m � 0.25 39. w � 4

41. x � �1 43. x � � (four roots) 45. x � �4, 39,596 47. x � 49. 13.1 in. by 9.1 in.

51. 1.65 ft or 3.65 ft 53. $30; 1,600 telephones

Exercise 1-8

1. �5 � x � 2 3. x � 3 or x 7 5. 0 � x � 8 7. �5 � x � 0 9. x � �2 or x 2(�5, 2) (�, 3) � (7, ) [0, 8] [�5, 0] (�, �2) � (2, )

11. �4 � x � 2 13. x � �4 or x 1 15. �5 � x � 0 or x 3 17. �3 � x � 1 19. x � 0 or x

(�4, 2] (�, �4] � (1, ) [�5, 0] � (3, ) (�3, 1) (�, 0) � ( , )

21. �4 � x � 23. �1 � x � 2 or x � 5 25. x � �4 or 0 � x � 2 27. x � �3 or x � 3 29. x � �2 or x �

(�4, ] (�1, 2) � [5, ) (�, �4] � [0, 2]

31. �7 � x � 3 33. If a 0, the solution set is (�, r1) � (r2 ). If a � 0, the solution set is (r1, r2).35. If a 0, the solution set is R, the set of real numbers. If a � 0, the solution set is {r}. 37. x2 � 0 39. No solution; 41. No solution; 43. x � 2 � or x � 2 � 45. 1 � � x � 0 or x 1 � 47. �2 � x � � or � x � 2 49. �2 � x � 2

(�, 2 � ] � [2 � , ) (1 � , 0) � (1 � , ) [�2, � ] � [ , 2] [�2, 2]

51. (A) Profit: $4 � p � $7 or ($4, $7) (B) Loss: $0 � p � $4 or p $7 or [$0, $4) � ($7, ) 53. 2 � t � 555. v 75 mph 57. 5 � t � 20

Chapter 1 Review Exercise*

1. x � 21 (1-1) 2. x � (1-1) 3. x � 3, y � 3 (1-2)4. x � 1 (1-3) 5. �14 � y � �4 (1-4) 6. �1 � x � 4 (1-4) 7. �5 � x � 4 (1-8) 8. x � �3 or x � 7 (1-8)

[1, ) (�14, �4) [�1, 4] (�5, 4) (�, �3] � [7, )

9. (A) 3 � 6i (B) 15 � 3i (C) 2 � i (1-5) 10. x � � or � (1-6) 11. x � 0, 2 (1-6)

12. x � , 3 (1-6) 13. m � � � ( /2)i (1-6) 14. y � (3 � )/4 (1-6) 15. x � 2, 3 (1-7)16. x � 3, y � �2 (1-2) 17. x � (1-3) 18. x � �15 (1-1) 19. No solution (1-1) 20. m � 2, n � � (1-2)

21. x � �19 (1-3) 22. x � 2 or x (1-4) 23. x � 0 or x (1-8) 24. x � 1 or 3 � x � 4 (1-8)

[�19, ) (�, 2) � ( , ) (�, 0) � ( , ) (�, 1] � (3, 4)

25. �1 � m � 2 (1-4) 26. �4 � x � 2 or [�4, 2) (1-8) 27. (A) 6 (B) 6 (1-4) 28. (A) 5 � 4i (B) �i (1-5)[�1, 2]

29. (A) �1 � i (B) (C) � 2i (1-5) 30. u � (�5 � )/2 (1-6) 31. u � 1 � i (1-6)

32. x � (1 � )/3 (1-6) 33. x � � , 64 (1-7) 34. m � �2, �3i (1-7) 35. y � , 3 (1-7) 36. x � 0.45 (1-1)37. �2.24 � x � 1.12 or [�2.24, 1.12] (1-3) 38. 0.89 � 0.32i (1-5) 39. x � �1.64, 0.89 (1-6)

94

278�43

�2�552

413 � 7

13i

m�1 2[ [

x3 41( ([x

0()12

x2

()103

] x�19

12

103

12

103

43

35

�33�312

12

12�14�7

2

x�3 7

[]( x�5 4([ x

�1 4[( y

�14 �4(] x

1

3011

[ x�2 2

[x[ [ [ [2�2 1

2�12

(( ( x1��2 0 1��2

[[ x2��5 2��5

12

12�2�2�5�5

12

12�2�2�5�5

x20�4

[] ]x[52�1

( ([ x�4

(32

32

32

32

x0

((14

( x�3 1((

30�5[ [ xx

�4 1([[ x

�4 2(

14

14

�2 2(( x[ x

�5 0[[ x

0 8[x

3 7((x

�5 2( (

� 4

5 � �17�5

�5 � �13

6

3i�3

232�2

�2

2

i�2

313

12

32

*The number in parentheses after each answer to a chapter review problem refers to the section in which that type of problem is discussed.

A-82 Answers

40. x � 0.94, y � 1.02 (1-2) 41. M � P/(1 � dt) (1-1) 42. I � (E � )/(2R) (1-6)43. y � (5 � x)/(2x � 4) (1-1) 44. The correct answer is x � �1. (1-1)45. If c � 9, there are two distinct real roots; if c � 9, there is one real double root; and if c 9, there are two distinct imaginary roots. (1-6)46. All real b and all negative a (1-3) 47. Less than 1 (1-3) 48. x � 1/(1 � y) (1-1)

49. 6 � d � x � 6 � d, x � 6 (1-4) 50. x � � i (1-6) 51. (four real roots) (1-7) 52. 1 (1-5)(6 � d, 6) � (6, 6 � d)

53. No solution (1-8) 54. Set of all real numbers (1-8)55. x � �4 or �2 � x � 0 or 0 � x � 2 or x � 4; (�, �4] � [�2, 0) � (0, 2] � [4, ) (1-8)56. u � �31 � 5x � 7y, v � 13 � 2x � 3y (1-2) 57. (A) Infinite number of solutions (B) No solution (1-2)58. or � (1-6) 59. (A) H � 0.7(220 � A) (B) 140 beats/min (C) 40 years old (1-1)60. 20 mL of 30% solution, 30 mL of 80% solution (1-1) 61. 3 mph (1-6)62. (A) 3 km (B) 16.2 km/h (C) 20.6 min (1-1, 1-6) 63. 85 bags of brand A and 45 bags of brand B (1-2)64. (A) 2,000 and 8,000 (B) 5,000 (1-6) 65. x � (13 � )/2 thousand, or approx. 3,146 and 9,854 (1-6)66. (13 � )/2 � x � (13 � )/2 or approx. 3.146 � x � 9.854, x in thousands (1-8) 67. �T � 110� � 5 (1-4)68. 20 cm by 24 cm (1-6) 69. 6.58 ft or 14.58 ft (1-7)

CHAPTER 2 Exercise 2-1

1. The y axis 3. Quadrant III 5. Quadrant IV 7. Quadrant II and quadrant IV 9. Quadrant I and quadrant IV11. 13. Symmetric with respect 15. Symmetric with respect 17. Symmetric with respect

to the origin to the x axis to the x axis, y axis, and origin

19. 21. 23. x2 � y2 � 16 25. (x � 3)2 � ( y � 2)2 � 1 27. (x � 2)2 � ( y � 6)2 � 329. (A) 3 (B) �2 (C) �3, �1, 4 (D) �4, 1, 331. (A) (B) (C) (D)

33. Symmetric with respect 35. Symmetric with respect 37. Symmetric with respect to 39. Symmetric with respect to to the x axis to the y axis the x axis, y axis, and origin the x axis, y axis, and origin

x

y

5

�5

�5 5x

y

2

�2

�2 2x

y

5

�5

�5 5x

y

5

�5

�5 5

x

y

5

5

�5

�5x

y

5

�5

5

�5x

y

5

�5

�5 5x

y

5

�5

�5

�82�106

x

y

5

�5

�5 5x

y

5

�5

�5 5x

y

5

�5

�5 5

x

y

5

�5

�5

5

�45�45�45

35

53

x6 � d 6 6 � d

( (

x � ��(2 � �3)/214

14�3

�E2 � 4PR

Answers A-91

39. 41. f �1(x) � 5x 43. f �1(x) � x � 45. f �1(x) � 5x � 2 47. f �1(x) � 2/(3 � x)

49. f �1(x) � x/(2 � x) 51. f �1(x) � (5x � 4)/(2 � x) 53. f �1(x) � 0.5 55. f �1(x) � (x � 2)5 �

57. f �1(x) � 9 � x2, x � 0 59. f �1(x) � �x2 � 4x � 1, x � 261. The x intercept of f is the y intercept of f�1, and the y intercept of f is the x intercept of f�1.63. f �1(x) � 65. f �1(x) �67. f �1(x) � 69. f �1(x) � � 71. f �1(x) � 73. f �1(x) � �

Domain of f �1 � [�3, 0] Domain of f �1 � [0, 3] Domain of f �1 � [1, 2] Domain of f �1 � [0, 1]Range of f �1 � [0, 3] Range of f �1 � [�3, 0] Range of f �1 � [0, 1] Range of f �1 � [�1, 0]

75. f �1(x) � (x � b)/a 77. a � 1 and b � 0 or a � �1 and b arbitrary 81. (A) f �1(x) � 2 � (B) f �1(x) � 2 �83. (A) f �1(x) � 2 � , 0 � x � 2 (B) f �1(x) � 2 � , 0 � x � 2

Chapter 2 Review Exercise

1. (A) (B) � (C) 2 (2-1, 2-2) 2. (A) x2 � y2 � 7 (B) (x � 3)2 � (y � 2)2 � 7 (2-1)3. Center: (�3, 2); radius � (2-1) 4. Slope � � (2-2) 5. 2x � 3y � 12 (2-2) 6. y � � x � 2 (2-2)

7. Vertical: x � �3, slope not defined; horizontal: y � 4, slope � 0 (2-2)8. (A) Function; domain � {1, 2, 3}, range � {1, 4, 9} (B) Not a function

(C) Function; domain � {�2, �1, 0, 1, 2}, range � {2} (2-3)9. (A) Not a function (B) A function (C) A function (D) Not a function (2-3)

10. Parts A and C specify functions. (2-3) 11. 16 (2-3) 12. 1 (2-3) 13. 3 (2-3) 14. �2a � h (2-3)15. 9 � 3x � x2 (2-5) 16. 1 � 3x � x2 (2-5) 17. 20 � 12x � 5x2 � 3x3 (2-5) 18. (3x � 5)/(4 � x2), x � �2 (2-5)19. 17 � 3x2 (2-5) 20. �21 � 30x � 9x2 (2-5)21. (A) (B) (C) (D)

(2-5)22. (A) g (B) m (C) n (D) f (2-4, 2-5)23. (A) x intercepts: �4, 0; y intercept: 0 (B) Vertex: (�2, �4) (C) Minimum: �4 (D) Range: y � �4 or [�4, )

(E) Increasing on [�2, ) (F) Decreasing on (�, �2] (2-4)24. Min f (x) � f (3) � 2; vertex: (3, 2) (2-4)25. (A) Reflected across x axis (B) Shifted down 3 units (C) Shifted left 3 units (2-5)

x

y

5

�5

�5 5x

y

5

�5

�5 5x

y

5

�5

�5 5x

y

5

�5

�5 5

x

y

�5 5

5

23

32�5

12�45

�4 � x2�4 � x2

�x�x

x

y

2

�2

�2 2

y � x

f �1

fx

y

2

�2

�2 2

y � x

f �1

f

x

y

5

�5

�5 5

y � x

f �1

f

x

y

5

�5

�5 5

y � xf �1

f

�2x � x2�2x � x2�9 � x2�9 � x2

�1 � �x � 31 � �x � 2

14

73

13�3 x � 5

72

12

10�10

�10

10

x

y

y � x

f

g

A-92 Answers

26. (A) 0 (B) 1 (C) 2 (D) 0 (2-4) 27. (A) �2, 0 (B) 1 (C) No solution (D) x � 3 and x � �2 (2-4)28. Domain � (�, ), range � (�3, ) (2-4) 29. [�2, �1], [1, ) (2-4) 30. [�1, 1) (2-4) 31. (�, �2) (2-4)32. x � �2, x � 1 (2-4) 33. f (x) � 4x3 � (2-3)34. The function f multiplies the square of the domain element by 3, adds 4 times the domain element, and then subtracts 6. (2-3)35. (A) 3x � 2y � �6 (B) (2-1, 2-2) 36. (A) y � �2x � 3 (B) y � x � 2 (2-2)37. It is symmetric with respect to all three. (2-1) 38. (�, 3) (2-3)39. Range � [�4, ) (2-4) 40. [0, 16) � (16, ) (2-3)

Intercepts: x � 1 and x � 5, y � 5Min f (x) � f (3) � �4

41. (A) ( f � g)(x) � , (g � f )(x) � (B) Domain of f � g � (�, ), domain of g � f � [0, ) (2-5)42. Functions (A), (C), and (D) are one-to-one (2-6) 43. (A) (x � 7)/3 (B) 4 (C) x (D) Increasing (2-6)44. Domain � [�1, 1] (2-4)

Range � [0, 1] � (2, 3]Discontinuous at x � 0

45. The graph of y � x2 is vertically expanded by a factor of 2, reflected in the x axis, and shifted to the left 3 units.Equation: y � �2(x � 3)2. (2-5)

46. g(x) � 5 � 3�x � 2� (2-5) 47. y � �(x � 4)2 � 3 (2-4, 2-5)

48. (A) (B) (C) (2-5)

x

y

5

�5

�5 5x

y

5

�5

x

y

5

�5

�5 5

x

y

5

5

x

y

2

4

�2 2

��x � 8��x� � 8

x

y

5

�5

5

Vertex(3, �4)

Axisx � 3

12�52

�x

Answers A-93

49. (A) f �1(x) � x2 � 1 50. (x � 3)2 � y2 � 32 (2-1) 51. Center: (�2, 3), radius � 4 (2-1)(B) Domain of f � [1, ) � Range of f �1

Range of f � [0, ) � Domain of f �1

(C) (2-6)

52. Symmetric with respect to the origin (2-1) 53. Decreasing (2-2, 2-3)

54. (A) Domain of f � [0, ) � Range of f �1; Range of f � [�1, ) � Domain of f �1 (B) (C) 2 (D) 4(E) x (2-6)

55. The graph of y � is vertically expanded by a factor of 2, reflected in the x axis, shifted 1 unit left and 1 unit down.Equation y � �2 � 1 (2-4)

56. It is the same as the graph of g shifted to the right 2 units, reflected in the x axis, and shifted down 1 unit. (2-5)57. (2-5) 58. [�5, 5] (1-8, 2-3)

59. (A) , domain � (�, 1] (B) , domain � (�, 1)(C) 1 � x, domain � (�, 1] (D) , domain � [�1, 1] (2-5)

60. (A) (3x � 2)/(x � 1) (B) (C) x (2-6)

61. (2-4)

62. x � y � 3; a line (2-1, 2-2) 65. (A) (B) (2-5)

66. Domain: All real numbers except x � 2, 67. (A) (B) (2-1, 2-5)Range: y �3 or (�3, ),discontinuous at x � 2 (2-4)

68. (A) The graph must cross the x axis exactly once. (B) The graph can cross the x axis either once or not at all. (2-4)�10

�10

10

10

x

y

5

�5

�5 5x

y

5

�5

�5 5

x

y

5

�5 5

x

y

5

�5 5

f (x) � ��2

2x

2

if x � �1

if �1 � x � 1; domain � (�, ), range � [�2, 2]

if x � 1

112

�1 � x2

x2/�1 � xx2�1 � x

x

y

�5

�5

�3 x � 1�3 x

�x � 1

x

y

5

5

x

y

5

�5

�5 5

y � x

f

f �1

A-94 Answers

69. (A) V � �1,250t � 12,000 (B) $5,750 (2-2) 70. (A) R � 1.6C (B) $168 (2-2)

71. ; E(2,000) � $200, E(5,000) � $400 (2-4)

72. (A) (B)

(C) 217 in 1995, 200 in 2000 (D) Per capita egg consumption is dropping about 17 eggs every 5 years. (2-4)

73. (A) (B) Discontinuous at x � 36 and x � 72 (2-4)

74. (A) C � 84,000 � 15x; R � 50x (B) R � C at x � 2,400 units: R � C for 0 � x � 2,400; R C for x 2,400 (2-2)

75. P( p) � �14,000 � 700p � 10p2 (2-5) 76. 5 ft (2-1)77. (A) A(x) � 60x � x2 (B) 0 � x � 40 (C) x � 20, y � 15 (2-4)78. (A) 0 (B) 1 (C) 2 (D) 0 (E) 1 (F) 0 (2-4)

Cumulative Review Exercise: Chapters 1 and 2

1. x � (1-1) 2. x � 1, y � �2 (1-2) 3. y � 5 (1-3) 4. �5 � x � 9 (1-4) 5. x � �5 or x � 2 (1-8)[5, ) (�5, 9) (�, �5] � [2, )

6. (A) 7 � 10i (B) 23 � 7i (C) 1 � i (1-5) 7. x � �4, 0 (1-6) 8. (1-6)9. (1-6) 10. x � 3 (1-7) 11. (1-3)

12. (A) (B) 2 (C) � (2-1, 2-2)13. (A) x2 � y2 � 2 (B) (x � 3)2 � (y � 1)2 � 2 (2-1) 14. Slope: ; y intercept: �2;

x intercept: 3 (2-2)

15. (A) Function: domain; {1, 2, 3}; range: {1} (B) Not a function(C) Function: domain: {�2, �1, 0, 1, 2}; range: {�1, 0, 2} (2-3)

x

y

5

�5

�5 5

23

122�5

x � �23 or [�2

3, )x � 3 � �7x � ��5, �5

x�5 2

[]( x�5 9(] y

5

52

32

R(x), C(x)

x

200,000

100,000

5,0002,5000

x

y

$50

$10

$20

$30

$40

10836 72

C(x) � �0.49x

0.44x

0.39x

for 00 � x � 36

for 36 � x � 72

for 72 � x

x

y

400

100

200

300

5 10 15 20

x

Consumption

f (x)

0

309

303

5

276

286

10

271

269

15

255

252

20

233

234

E(x) � � 200

0.1x � 100

if 0 � x � 3,000

if x 3,000

A-100 Answers

53. 55. 57.

59. p(x) � x2 � 1, [ f(x) � p(x)] → 0 as x → � 61. p(x) � x3 � x, [ f(x) � p(x)] → 0 as x → �

63. Domain: x � 2, or (�, 2) � (2, ); 65. Domain: x � 2, �2, 67. N → 50 as t →

f(x) � x � 2 or (�, �2) � (�2, 2) � (2, );

f(x) �

69. N → 5 as t → 71. (A) 73. (A)

(B) 10 yr (B) (0, )(C) (C) 15 ft by 15 ft

(D)

Exercise 3-5

1. A � 2, B � 3 3. A � 3, B � �5 5. A � 2, B � 3, C � �4 7. A � 3, B � �2, C � 5

9. A � 2, B � �1, C � �3, D � 0 11. 13. 15.

17. 19. 21. 23.

25. 27. 29.


1. 2x3 � 3x2 � 1 � (x � 2)(2x2 � x � 2) � 5 (3-1) 2. P(3) � �8 (3-1, 3-2) 3. 2, �4, �1 (3-2) 4. 1 � i (3-3)5. (A) P(x) � (x � 2)x(x � 2) � x3 � 4x (B) P(x) → as x → and P(x) → � as x → � (3-1)6. Lower bound: �2, �1; upper bound: 4 (3-3) 7. P(1) � �5 and P(2) � 1 are of opposite sign. (3-3)8. �1, �2, �3, �6 (3-2) 9. �1, 2, 3 (3-2)

10. (A) Domain: (�, �4) � (�4, ); x intercept: (B) Domain: (�, �2) � (�2, 3) � (3, ); x intercept: 0 (3-4)11. (A) Horizontal asymptote: y � 2; vertical asymptote: x � �4

(B) Horizontal asymptote: y � 0; vertical asymptotes: x � �2, x � 3 (3-4)

12. (3-5)2

x � 3 �

5

x � 2

32

1

(a � b)(x � a) �

1

(a � b)(x � b)

1

x � a �

a

(x � a)22x � 7

x � 3 �

1

x � 1 �

3

x2 � 2x � 3

x � 2 � 3

x � 2 �

2

(x � 2)2 � x � 6

x2 � 2x � 4

3

x � 1 �

x � 3

x2 � x � 2

4x � 5

x2 � 1 �

2x

(x2 � 1)2

3

x �

4x � 1

2x2 � x � 1

4

x �

5

2x � 1 �

3

(2x � 1)2

4

2x � 7 �

2

3x � 2

�4

x � 4 �

7

x � 3

x

L

100

200

50250

n

C

1,000

2,000

50250

t

N

50

25

50250

L(x) � 2x � 450

xC(n) � 25n � 175 �

2,500

n

x

y

5

�5

5

1

x � 2

x

y

5

�5

�5 5

t

N

50

25

50250

x

y

�5

�5 5x

y

5

�5

�5 5x

y

5

�5

�5 5

Answers A-101

13. (A) The graph of P(x) has three x intercepts and (B) 3.5 (3-1, 3-3)two turning points; P(x) → as x → andP(x) → � as x → �

14. Q(x) � 8x3 � 12x2 � 16x � 8, R � 5; P( ) � R � 5 (3-1) 15. �4 (3-1) 16. P(x) � [x � (1 � )][x � (1 � )] (3-2)17. Yes, since P(�1) � 0, x � (�1) � x � 1 must be a factor. (3-2) 18. �2, � , 4 (3-2)

19. P(x) � (x � 2)(2x � 1)(x � 4) (3-2) 20. No rational zeros (3-2) 21. �1, , (3-2)

22. (3-2)

23. Inequality notation: x � �3 or � � x � 2; interval notation (�, �3] � [� , 2] (3-2, 3-8)24. (A) Upper bound: 7; lower bound: �5 (B) 6.62 (C) �4.67, 6.62 (3-3)25. (A) Domain: (�, �1) � (�1, ); x intercept: 1; y intercept: � (B) Vertical asymptote: x � �1; horizontal asymptote: y �

(C) (3-4)

26. (3-5) 27. (3-5)

28. P(x) � [x � (1 � i)][x2 � (1 � i)x � (3 � 2i)] � (3 � 5i) (3-1) 29. P(x) � (x � )2(x � 3)(x � 1)3, degree 6 (3-2)

30. P(x) � (x � 5)[x � (2 � 3i)][x � (2 � 3i)], degree 3 (3-2) 31. , �2, 1 � (3-2)

32. (x � 2)(x � 2)(2x � 1)[x � (1 � )][x � (1 � )] (3-2)

33. Inequality notation: �3 � x � � or � � x � or x 2; interval notation: (�3, � ] � (� , ] � (2, ) (3-2, 3-8)34. Since P(x) changes sign three times, the minimal degree is 3. (3-3)35. P(x) � a(x � r)(x2 � 2x � 5), and since the constant term, �5ar, must be an integer, r must be a rational number. (3-2)

36. (A) 3 (B) (3-2) 37. (A) Upper bound: 30; lower bound: �30 (B) �23.54, 21.57 (3-3)

38. (3-4)

39. y � 2 and y � �2 (3-4) 40. (3-5) 41. 2x3 � 32x � 48 � 0, 4 � 12 ft or 5.2 � 9.2 ft (3-2)

42. x3 � 27x2 � 729 � 0, 4.8 ft (3-3) 43. 4x3 � 70x2 � 300x � 300, 1.4 in. or 4.5 in. (3-3)44. x4 � 7x2 � 2x � 8, (�2, 4), (�1.6, 2.6), (1, 1), (2.6, 6.8) (3-2)

2

x � 3 �

3

x �

x � 1

x2 � 1

x

y

5

�5

�5

5

�3

2 �

3i�3

2

12

12

32

12

12

32

�2�2�21

2

12

3

x �

2x � 1

2x2 � 3x � 3

1

x �

2

x � 2 �

3

(x � 2)2

x

y

5

�5

�5

5

12

12

12

12

(x � 1)(2x � 1)�x � 1 � i�3

2 ��x � 1 � i�3

2 �

1 � i�3

212

12

�2�214

P(x)

x

10

�10

�5 5

Answers A-105

Exercise 4-4

1. 4.4408 3. �2.3644 5. �7.3324 7. 6.1242 9. 32.45 11. 0.1039 13. 0.055 68 15. 3,407 17. 1.23819. 2.320 21. �51.083 23. 5.192 25. 35.779 27. �12.169 29. 4.6505 � 1021 31. 1.4925 � 10�5

33. 35. 37. 39.

41. The inequality sign in the last step reverses because log is negative. 43. (B) Domain � (1, ); range � (�, )45. (0.90, �0.11), (38.51, 3.65) 47. (6.41, 1.86), (93.35, 4.54) 49. 51.

53. (A) 0 decibel (B) 120 decibels 55. 30 decibels more 57. 8.6 59. 1,000 times more powerful 61. 7.67 km/s63. (A) 8.3, basic (B) 3.0, acid 65. 6.3 � 10�6 mole/liter

Exercise 4-5

1. 1.44 3. 5.88 5. �1.68 7. 0.0967 9. 3.24 11. �3.43 13. 15. 8/e2 17. x � 10 19. 86.721. �921 23. 5.18 25. �4.01 27. x � 5 29. 31. 33. x � 1, e2, e�2

35. x � ee 37. x � 0.1, 100 39. (B) 2 41. (B) �1.252, 1.707 43. 3.6776 45. �1.6094 47. �1.7372

49. 51. I � I0(10D/10) 53. I � I0[10(6�M )/2.5] 55. 57.

59. 61. 63. 65. 0.38 67. 0.55 69. 0.57

71. 0.85 73. 0.43 75. 0.27 77. Approx. 5 years 79. 9.16% 81. (A) 6 (B) 100 times brighter83. Approx. 35 years 85. 18,600 years old 87. 7.52 s 89. k � 0.40; t � 2.9 h 91. 10 years


1. log m � n (4-3) 2. ln x � y (4-3) 3. x � 10 y (4-3) 4. y � ex (4-3) 5. 72x (4-1) 6. (4-1)7. x � 8 (4-3) 8. x � 5 (4-3) 9. x � 3 (4-3) 10. x � 1.24 (4-3) 11. x � 11.9 (4-3) 12. x � 0.984 (4-3)

13. x � 103 (4-3) 14. x � 4 (4-3) 15. x � 2 (4-3) 16. x � �1, 3 (4-2) 17. x � 1 (4-1) 18. x � �3 (4-2)19. x � �2 (4-3) 20. x � (4-3) 21. x � 64 (4-3) 22. x � e (4-3) 23. x � 33 (4-3) 24. x � 1 (4-3)25. 1.145 (4-3) 26. Not defined (4-3) 27. 2.211 (4-3) 28. 11.59 (4-3) 29. x � 41.8 (4-1)30. x � 1.95 (4-3) 31. x � 0.0400 (4-3) 32. x � �6.67 (4-3) 33. x � 1.66 (4-3) 34. x � 2.32 (4-5)35. x � 3.92 (4-5) 36. x � 92.1 (4-5) 37. x � 2.11 (4-5) 38. x � 0.881 (4-5) 39. x � 300 (4-5)40. x � 2 (4-5) 41. x � 1 (4-5) 42. (4-5) 43. x � 1, 103, 10�3 (4-5) 44. x � 10e (4-5)45. e�x � 1 (4-2) 46. 2 � 2e�2x (4-2)

x � 12(3 � �13)

13

e2x2

�10

�10

10

10

�10

�10

10

10

x � 1

2 ln

1 � y

1 � y

x � ln (y � �y2 � 1)t � �L

R ln �1 �

RI

E �r � ln (A/P)

t

x � 14(1 � �89)x � 2 � �3

52

�2

�1

2

3

�2

�1

2

3

13

x

y

�10

�5

5

5 10

x

y

5

�5

�5 5x

y

�5

5

5 10x

y

�5

5

5 10

A-106 Answers

47. (4-1) 48. (4-2) 49. (4-3)

50. (4-2) 51. y � �ex; y � (4-3)

52. (A) y � e�x/3 is decreasing, while y � 4 ln (x � 1) is increasing without bound. (B) 0.258 (4-5) 53. 0.018, 2.187 (4-3)

54. (1.003, 0.010), (3.653, 4.502) (4-4) 55. I � I0(10D/10) (4-5) 56. (4-5) 57. I � I0(e�kx) (4-5)

58. (4-5) 59. f �1(x) � ex/2 � 1 (4-5, 1-7) 60. f �1(x) � (4-5, 2-6)

61. y � ce�5t (4-3, 4-5)62. Domain f � (0, ) � Range f �1

Range f � (�, ) � Domain f �1 (4-3)

63. If log1 x � y, then we have 1y � x; that is, 1 � x for arbitrary positive x, which is impossible. (4-3) 65. 23.4 years (4-5)66. 23.1 years (4-5) 67. 37,100 years (4-5) 68. (A) N � 22t or N � 4t (B) 15 days (4-5) 69. $1.1 � 1026 (4-2)70. (A) (B) 0 (4-2) 71. 6.6 (4-4) 72. 7.08 � 1016 joules (4-4)

73. 50 decibels more (4-4) 74. k � 0.009 42; 489 ft (4-2) 75. 3 years (4-5)


1. (A) P(x) � (x � 1)2(x � 1)(x � 2) (B) P(x) → as x → and as x → � (3-1)2. 3x3 � 5x2 � 18x � 3 � (x � 3)(3x2 � 4x � 6) � 15 (3-1) 3. �2, 3, 5 (3-2)4. P(1) � �5 and P(2) � 5 are of opposite sign (3-3) 5. 1, 2, �4 (3-2)

6. (3-5) 7. (A) x � log y (B) x � ey (4-4) 8. (A) 8e3x (B) e5x (4-2)

9. (A) 9 (B) 4 (C) (4-3) 10. (A) 0.371 (B) 11.4 (C) 0.0562 (D) 15.6 (4-4)11. (4-3)12. The function f multiplies the base e raised to a power the domain element by 100 and then subtracts 50. (4-2)13. (3-2) 14. Part B (3-1)P(1

2) � 52

12

f (x) � 3 ln x � �x

12

3

x � 1�

2

x � 2

t

P

1,000

500

50250

x

y

f �1: y � 2x

f : y � log2x

10

5

�5

�5 105

ln (x � �x2 � 1)n � �ln [1 � (Pi/r)]

ln (1 � i)

x � ��2 ln (�2�y)

�1e�x

� e�x

t

N

50

100

�5 5

x

y

5

�5

5 10

t

f

50

100

�20 20

x

y

5

10

�5 5

A-114 Answers

53. (A) (B) The domain of cos�1 is restricted to �1 � x � 1; hence no graph will appear for other x.

55. 57. 1/2 59. 61. 63. f �1(x) � 3 � cos�1 [(x � 4)/2], 2 � x � 665. (A) (B) The domain for cos x is (�, ), and the range is [�1, 1], which is the domain for cos�1 x.

Thus, y � cos�1 (cos x) has a graph over the interval (�, ), but cos�1 (cos x) � x only onthe restricted domain of cos x, [0, �].

67. 75.38°; 24.41°69. (A) (B) 59.44 mm 71. 21.59 in.

73. (A) (B) 7.22 in. 75. (B) 76.10 ft


1. 2.5 radians (5-1) 2. 7.5 cm (5-1) 3. � � 54.8°, a � 16.5 ft, b � 11.6 ft (5-2)4. (A) �/3 (B) 60° (C) �/6 (D) 30° (5-2, 5-4) 5. (A) III, IV (B) II, III (C) II, IV (5-3)6. (A) � (B) (C) � (5-3)7. (5-4) 8. (A) 2� (B) 2� (C) � (5-6)

9. (A) Domain � (�, ), range � [�1, 1]

(B) Domain is the set of all real numbers except x � �, k an integer, range is the set of all real numbers (5-6)

10. (5-6) 11. (5-6)

x

y

5

��

x

y

1

�1

2�

��2� ��

2k � 1

2

030456090

180270360

�°

0�/6�/4�/3�/2�

3�/22�

� rad

01/21/��2��3/210�10

sin �

1��3/21/��21/20�101

cos �

01/��31��3ND0ND0

tan �

ND*2��22/��31ND�1ND

csc �

12/��3��22ND�1ND1

sec �

ND��311/��30ND0ND

cot �

*ND � not defined.

43

54

35

0

3

35

10

0

3

35

10

0

10

150

100

0

10

150

100

0

�2�

�

2�0

0

�

�

1/�1 � x2�1 � x2�2/2�1

�2

1

2

�1

�1

1

1

Answers A-115

12. The central angle in a circle subtended by an arc of half the length of the radius. (5-1)13. If the graph of y � sin x is shifted �/2 units to the left, the result will be the graph of y � cos x. (5-6, 5-7) 14. 78.50° (5-1)15. � � 49.7°, � � 40.3°, c � 20.6 cm (5-2) 16. (A) II (B) Quadrantal (C) III (5-1) 17. (A) and (C) (5-1)18. (B) and (C) (5-3, 5-5) 19. (A) �/2, 3�/2 (B) 0, � (C) 0, � (5-3, 5-5)20. Since the coordinates of a point on a unit circle are given by P(a, b) � P(cos x, sin x), we evaluate P(cos (�8.305), sin (�8.305))—using

a calculator set in radian mode—to obtain P(�0.436, �0.900). Note that x � �8.305, since P is moving clockwise. The quadrant inwhich P(a, b) lies can be determined by the signs of a and b. In this case P is in the third quadrant, since a is negative and b is negative.(5-5)

21. � (5-4) 22. 1/ or /2 (5-4) 23. �1 (5-4) 24. 1/ or /3 (5-4) 25. �/2 (5-4, 5-9)26. ��/2 (5-4, 5-9) 27. �/4 (5-4, 5-9) 28. ��/3 (5-4, 5-9) 29. Not defined (5-4, 5-9) 30. 2�/3 (5-4, 5-9)31. �/3 (5-4, 5-9) 32. Not defined (5-4, 5-9) 33. (5-4, 5-9) 34. 5�/6 (5-4, 5-9) 35. (5-4, 5-9)36. (5-4, 5-9) 37. 0.4431 (5-3) 38. �15.17 (5-3) 39. �2.077 (5-3, 5-5) 40. �0.9750 (5-2, 5-9)41. Not defined (5-2, 5-9) 42. 1.557 (5-2, 5-9) 43. 1.095 (5-9) 44. Not defined (5-9)45. (A) � � �30° (B) � � 120° (5-9) 46. (A) � � 151.20° (B) � � 82.28° (5-9)47. cos�1 [cos (�2)] � 2 For the identity cos�1 (cos x) � x to hold, x must be in the restricted domain of the cosine function; that is,

0 � x � �. The number �2 is not in the restricted domain. (5-9)48. A � 2, P � 2 (5-7) 49. (5-7) 50. y � 6 cos 2x, ��/2 � x � � (5-7)

51. y � �0.5 sin �x, �1 � x � 2 (5-7)52. If the graph of y � tan x is shifted �/2 units to the right and reflected in the x axis, the result will be the graph of y � cot x. (5-6, 5-7)

53. (A) cos x (B) tan2 x (5-5) 54. (5-7) 55. A � 2, P � 4, phase shift � (5-7)

56. y � cos�1 x � arccos x (5-9) 57. (5-7)Domain � [�1, 1]Range � [0, �]

58. (A) y � tan x (B) y � cot x(5-8)

59. (A) 2.5 rad (B) (�6.41, 4.79) (5-1, 5-3) 60. (A) 2�/3 (B) 5�/4 (5-5)�5

�2�

5

2�

�5

�2�

5

2�

0

�2�

2

2�

x

y

1

�

�1

�2�0, �

�2

(�1, �)

(1, 0)

y � 12 � 12 cos 2x

12

x

y

�3

3

4�2��2��4�

y

3

�5

4�

2��2��4�x

x

y

2

�2

321�1

�7/4�52

3

�3�3�2�212

A-116 Answers

61. (5-6) 62. y � tan�1 x � arctan x (5-9) 63. P � 1; phase shift � � (5-8)Domain � (�, )Range � (��/2, �/2)

64. Period � 4�; phase shift � �/2 (5-8) 65. (A) Origin (B) y axis (C) Origin (5-6) 66. 1/ (5-9)67. For each case, the number is not in the domain of the function and an error message of some type will appear. (5-5, 5-9)68. y � 2 sin (�x � �/4) (5-7) 69. True (5-3) 70. True (5-4) 71. False (5-4) 72. False (5-6)73. True (5-6) 74. False (5-9) 75. False (5-7) 76. True (5-7) 77. Yes (5-1, 5-6)83. y � 2 sin (2x � 0.928) (5-7) 84. (A) (B) (5-7)

85. 2�/5 rad (5-1) 86. 28.3 cm (5-2) 87. I � 30 cos 120�t (5-7)88. (A) L � 10 csc � � 15 sec �, 0 � � � �/2

(B) Length of longest log that can make the corner is 35 ft.

(C) Length of longest log that can make the corner is 35.1 ft.

(D) The length L increases without end. (5-2, 5-5)89. (A) y � 4 � 3 cos (�t/6)

(B) The graph shows the seasonal changes in soft-drink consumption. Most is consumed in August and the least in February. (5-7)90. (A) (B) y � 66.5 � 8.5 sin [(�x/6) � 2.4] (C) (5-7)


25. 27. 29. No 31. Yes 33. No 35. No

65. Not an identity 67. An identity 69. Not an identity 71. An identity 73. An identity 75. Not an identity�2

��

2

�

�2

��

2

�

40

1

90

24

40

1

90

24

30

0.4

40

1.0

� (rad)

L (ft)

0.4

42.0

0.5

38.0

0.6

35.9

0.7

35.1

0.8

35.5

0.9

36.9

1.0

39.6

TABLE 2

�3

�2�

3

2� x

y

2

1

�2

3�

�3�

�2

�3�

2

3�

�1 � x2

x

y

1

�1

�2

�2�

�4��1, � �

�4�1, �

12

x

y

5

�5

�

�2�

�2

3�2

Answers A-119

49. (B)

Exercise 6-5

1. 7�/6, 11�/6 3. 7�/6 � 2k�, 11�/6 � 2k�, k any integer 5. 2�/3 7. 2�/3 � k�, k any integer 9. 30°, 330°11. 1.1279, 5.1553 13. 74.0546° 15. 3.5075 � 2k�, 5.9172 � 2k�, k any integer 17. 0.3376 19. 2.764221. k(180°), 135° � k(180°), k any integer 23. 0, 2�/3, �, 4�/3 25. 210°, 330° 27. 60°, 180°, 300° 29. �/3, �, 5�/331. 41.81° 33. 1.911 35. 0.3747, 2.767 37. Infinitely many 39. One 41. Infinitely many 43. 0.3747, 2.766945. 0.3747 � 2k�, 2.7669 � 2k�, k any integer 47. (�1.1530, 1.1530) 49. [3.5424, 5.3778], [5.9227, ) 51. 1.818353. tan�1 (�5.377) has exactly one value, �1.387; the equation tan x � �5.377 has infinitely many solutions, which are found by adding k�,

k any integer, to each solution in one period of tan x.55. 0, 3�/2 57. � 59. 0.1204, 0.138461. (A) The largest zero for f is 0.3183. As x increases without bound, 1/x tends to 0 through positive numbers, and sin (1/x) tends to 0

through positive numbers. y � 0 is a horizontal asymptote for the graph of f.(B) Infinitely many zeros exist between 0 and b, for any b, however small. The exploration graphs suggest this conclusion, which is

reinforced by the following reasoning: Note that for each interval (0, b], however small, as x tends to zero through positive numbers,1/x increases without bound, and as 1/x increases without bound, sin (1/x) will cross the x axis an unlimited number of times. Thefunction f does not have a smallest zero, because, between 0 and b, no matter how small b is, there is always an unlimited numberof zeros.

63. 0.009235 s 65. 50.77° 67. 123° 69. 2.267 rad 71. (A) 12.4575 mm (B) 2.6496 mm73. (r, �) � (0, 0), (0, 180°), (0, 360°) 75. � � 67°

Chapter 6 Review Exercises

5. (cos 5� � cos �) (6-4) 6. 2 cos 7x sin 2x (6-4) 7. cos x (6-2)8. 135° � k(360°), 225° � k(360°), k any integer (6-5) 9. k�, �/4 � k�, k any integer (6-5)

10. �34.7648° � k(360°), k any integer (6-5) 11. �0.0065 (6-5) 12. 0.5943 � 2k�, 2.5473 � 2k�, k any integer (6-5)

13. 3.1855 (6-5) 14. (A) Not an identity (B) An identity (6-1) 24. (6-4) 25. (6-4)

26. No (6-1) 27. Yes (6-3) 28. Yes (6-2) 29. No (6-2) 30. �/3, 2�/3, 4�/3, 5�/3 (6-5) 31. 0°, 120° (6-5)32. k�, �/6 � 2k�, 5�/6 � 2k�, k any integer (6-5) 33. k�, �/6 � 2k�, 11�/6 � 2k�, k any integer (6-5)34. 120° � k(360°), 240° � k(360°), k any integer (6-5) 35. 14.34° � k(180°), k any integer (6-5)36. 0.6259 � 2k�, 2.516 � 2k�, k any integer (6-5) 37. 1.178, 2.749 (6-5) 38. Two (6-5) 39. Infinitely many (6-5)40. None (6-5) 41. Infinitely many (6-5) 42. 1.4903 (6-5) 43. x � 1.4903 (6-5) 44. �0.6716, 0.6716 (6-5)45. [�0.6716, 0.6716] (6-5)46. (A) Yes (B) Conditional equation, since the equation is false for x � 1 and y � 1, for example, and both sides are defined at x � 1

and y � 1. (6-1)47. sin�1 0.3351 has exactly one value, and the equation sin x � 0.3351 has infinitely many solutions. (5-9, 6-5)48. (A) Not an identity (B) An identity (6-1)

49. (6-2)

50. (A) 0, 2�/3, 4�/3 (B) 0, 2.0944, 4.1888 (6-5) 51. �2.233, 0.149 (6-5)52. (A) 3/ or 3 /10 (B) 7/25 (6-3) 53. �24/25 (6-3) 54. 24/25 (6-2)�10�10

�2

�2�

2

2�

y2 � 12 cos x � �3

2 sin x

��6

2

�2 � �3

4

12

�1

0

1

0.25

�1

0

1

0.25

�1

0

1

0.25

�1

0

1

0.25

A-120 Answers

55. (A) 0, �/3, 2�/3 (B) 0, 1.0472, 2.0944 (6-5)56. (A) 0.6817, 1.3183

(B) As x increases without bound, 1/(x � 1) tends to 0 through positive numbers, and sin [1/(x � 1)] tends to 0 through positive num-bers. y � 0 is a horizontal asymptote for the graph of f.

(C) There are infinitely many zeros in any interval containing x � 1. The number x � 1 is not a zero because sin [1/(x � 1)] is notdefined at x � 1. (6-5)

57. x � ; x � 5.196 cm, � � 30.000° (6-3) 58. 0.00346 s (6-5)59. y � 0.6 cos (184�t) y � �0.6 cos (208�t)

y � 0.6 cos (184�t) � 0.6 cos (208�t) y � 1.2 sin (12�t) sin (196�t)(6-4)

60. Height � 7.057 ft, radius � 21.668 ft

From the figure, R� � 18 and sin � � 16/R. From these two equations, solving each for R in terms of � andsetting the results equal to each other, we obtain the desired trigonometric equation. (6-5)


1. � � 79°, a � 41 ft, b � 20 ft 3. � � 40°, a � 16 km, c � 5.8 km 5. � � 49°, a � 53 yd, b � 66 yd7. � � 81°, b � 16 cm, c � 12 cm 9. Two triangles; case (c) 11. One triangle; case (b) 13. Zero triangles; case (a)

15. Zero triangles; case (e) 17. One triangle; case (d) 19. One triangle; none of the cases21. � � 49.5°, a � 20.0 ft, c � 4.81 ft 23. � � 58.1°, a � 140 m, c � 129 m 25. No solution27. Triangle 1: � � 63.4°, � � 77.7°, c � 46.7 in.; Triangle 2: � � 116.6°, � � 24.5°, c � 19.8 in. 29. No solution31. � � 22°10�, � � 128°20�, c � 89.8 mm 33. k � 25.2 sin 42.3° � 16.9599 35. Left side: 16.204; right side: 16.07337. 4.06 mi, 2.47 mi 39. 353 ft 41. 5.8 in., 3.1 in. 43. 4.42 � 107 km, 2.39 � 108 km 45. 159 ft47. R � 7.76 mm, s � 13.4 mm

Exercise 7-2

1. Angle � is acute. A triangle can have at most one obtuse angle. Since � is acute, then, if the triangle has an obtuse angle, it must be theangle opposite the longer of the two sides, b and c. Thus, �, the angle opposite the shorter of the two sides, c, must be acute.

3. a � 6.03 yd, � � 56.6°, � � 52.2° 5. c � 14.0 mm, � � 20°40�, � � 39°0�

7. If the triangle has an obtuse angle, then it must be the angle opposite the longest side; in this case, �.9. � � 23.0°, � � 94.9°, � � 62.1° 11. � � 67.3°, � � 54.6°, � � 58.1° 13. No solution

15. b � 23.5 inches, � � 28.3°, � � 25.5° 17. No solution 19. � � 30.7°, � � 110.9°, c � 21.0 in.21. � � 49.1°, � � 102.9°, � � 28.0°23. Triangle 1: � � 70.3°, � � 51.3°, a � 5.99 m; Triangle 2: � � 109.7°, � � 11.9°, a � 1.58 m 25. No solution 33. 120 yd35. 5.81 ft 37. 121 mi 39. 74.1 m 41. 0.284 rad 43. � � 31°50�, � � 50°10�, � � 98°0� 45. �CAB � 33°47. 24,800 mi

Exercise 7-3

1. �u � v� � 78 mph, � � 67° 3. �u � v� � 41 kg, � � 45° 5. �u� � 12 lb, �v� � 21 lb 7. �u� � 388 mph, �v� � 41 mph9. �u � v� � 77g, � � 15° 11. �u � v� � 23 knots, � � 6° 13. �u� � 12 kg, �v� � 6.0 kg

18 18

16 16h

RR � �

�2

0.0

2

0.2

�2

0.0

2

0.2

�2

0.0

2

0.2

�2

0.0

2

0.2

�27

A-124 Answers

25. True 27. False29. (A) (1 � i)4 � 4 � �4 � 4 � 0; there are three other roots.

(B) The four roots are equally spaced around the circle. Since there are four roots, the angle between successive roots on the circle is360°/4 � 90°.

(C) (�1 � i)4 � 4 � �4 � 4 � 0; (�1 � i)4 � 4 � �4 � 4 � 0; (1 � i)4 � 4 � �4 � 4 � 031. x1 � 4e60°i � 2 � 2 i, x2 � 4e180°i � �4, x3 � 4e300°i � 2 � 2 i

33. x1 � 3e0°i � 3, x2 � 3e120°i � , x3 � 3e240°i � 35. w1 � 2i, w2 � �2i

37. w1 � � i, w2 � � � i, w3 � �2i 41. x1 � 2e0°i, x2 � 2e72°i, x3 � 2e144°i, x4 � 2e216°i, x5 � 2e288°i

43. x1 � e36°i, x2 � e108°i, x3 � e180°i, x4 � e252°i, x5 � e324°i

45. P(x) � (x � 2i)(x � 2i)[x � (� � i)][x � (� � i)][x � ( � i)][x � ( � i)]


1. 1 (7-1) 2. 0 (7-1) 3. 2 (7-1)4. Angle � is acute. A triangle can have at most one obtuse angle. Since � is acute, then, if the triangle has an obtuse angle, it must be the

angle opposite the longer of the two sides, b and c. Thus, �, the angle opposite the shorter of the two sides, b, must be acute. (7-2)5. � � 75°, a � 47 m, b � 31 m (7-1) 6. a � 4.0 ft, � � 36°, � � 129° (7-1, 7-2) 7. � � 40°, � � 19°, a � 8.2 cm (7-1)8. � � 19°, �u � v� � 170 mi/h (7-3) 9. 3, �7� (7-4) 10. (7-4)

11. (7-5) 12. (7-5) 13. (7-6)

14. (�10, �210°): The polar axis is rotated 210° clockwise (negative direction), and the point is located 10 units from the pole along thenegative polar axis. (�10, 150°): The polar axis is rotated 150° counterclockwise (positive direction), and the point is located 10 unitsfrom the pole along the negative polar axis. (10, 330°): The polar axis is rotated 330° counterclockwise, and the point is located 10 unitsfrom the pole along the positive polar axis. (7-5)

15. (7-6) 16. (A) 2e(�60°)i (B) 2 � 2i (7-6) 17. (A) 1 (B) 1 (7-7)

18. 8 � i8 (7-7) 19. If the triangle has an obtuse angle, then it must be the angle opposite the longest side; in this case, �. (7-2)20. b � 10.5 cm, � � 27.2°, � � 37.4° (7-2) 21. No solution (7-1)22. Two solutions. Obtuse case: � � 133.9°, � � 19.7°, c � 39.6 km (7-1) 23. � � 41.1°, � � 74.1°, � � 64.8° (7-1, 7-2)24. The sum of all of the force vectors must be the zero vector for the object to remain at rest. (7-4)

�3

�3

5 100

�2

3�2

�

CA

B

x

y

5

�5

�5 5

A

B

C5 100

�2

3�2

�5 10

� � �6

0

�2

3�2

�

�34

�3�3�3�3

�3�3

�3

2 �

3�3

2i�

3

2 �

3�3

2i

�3�3

x

y

�1 � i

�1 � i

1 � i

1 � i

radius �2

Answers A-125

25. �u � v� � 98.0 kg, � � 17.1° (7-3) 26. u � 11i � 7j (7-4)

27. 105, 45� (7-4) 28. u � (7-4) 29. k � (7-4) 30. k1 � , k2 � (7-4) 31. (7-4)

32. (7-5) 33. (7-5) 34. (7-5)

35. (7-5) 36. (7-5) 37. (7-5)

38. n � 1 n � 2 n � 3 Two leaves for all n (7-5)

39. (A) Ellipse (B) Parabola (C) Hyperbola (7-5)

40. r2 � 6r cos � or r � 6 cos � (7-5) 41. x2 � y2 � 5x (7-6)42. z1 � e135°i, z2 � 2e(�120°)i, z3 � 5e0°i (7-6) 43. z1 � 1 � i, z2 � (�3 /2) � i, z3 � �1 � i (7-6)

44. (A) 32e44°i (B) 2e6°i (7-6) 45. (A) �8 � 8 i (B) �8 � 13.86i (7-7)

46. (7-7)

x

y

w2

w3

w1

w3 � �i

w2 � ��3

2 �

1

2i

w1 � �3

2 �

1

2i

�3�33

2�3�2

�10

�10

10

10

�10

�10

10

10

�10

�10

10

10

�10

�10

10

10

�10

�10

10

10

�10

�10

10

10

�10

�10

10

10

�10

�10

10

10

5 100

�2

3�2

�

50

�2

3�2

�10 20

0

�2

3�2

�5

0

�2

3�2

�

�15

13, �

36

13�2

5

1

5

6 � �6

10��4

5,

3

5�

A-126 Answers

47. 2e50°i, 2e170°i, 2e290°i (7-7) 48. (4e15°i)2 � 16e30°i � 8 � 8i (7-7)49. (5.76, �26.08°) (7-5) 50. (�5.30, �2.38) (7-5) 51. 5.26e127.20°i (7-6) 52. �7.27 �2.32i (7-6)53. (A) There are a total of three cube roots, and they are spaced equally around a circle of radius 2.

(B) w2 � � � i, w3 � � i (C) The cube of each cube root is �8i. (7-7)54. k � 44.6 sin 23.4° (7-1) 57. False (7-7) 58. False (7-7) 59. True (7-7) 60. False (7-7)61. (A) (B) (7-5)

62. (A) The coordinates of P represent a simultaneous solution.

(B) r � �4 , � � 3�/4 (C) The two graphs go through the pole at different values of �. (7-5)63. 1, �1, i, �i, �2 � i �2, �2 � i �2, � �2 � i �2, � �2 � i �2 (7-7)64. P(x) � (x � 2i)[x � (� � i)][x � ( � i)] (7-7) 65. 438 mi (7-3) 66. 438 mph at 83° (7-3)67. 86°, 464 mph (7-3) 68. 0.6 mi (7-1) 69. 177 lb at 15.2° relative to v (7-3) 70. 19 kg at 204° relative to v (7-4)71. (A) Distance at aphelion: 1.56 � 108 mi (B) Distance at aphelion: 1.56 � 108 mi

Distance at perihelion: 1.29 � 108 mi Distance at perihelion: 1.29 � 108 mi (7-5)

72. 5,740 lb (7-4)

Cumulative Review Exercise Chapters 5–7

1. 1.86 m (5-3) 2. � � 57.3°, 14.5 cm, 7.83 cm (5-5) 3. (A) I, II (B) I, IV (C) I, III (5-4)4. (A) � (B) (C) � (5-4) 5. (A) �/4 (B) 65° (C) 30° (5-4)6. (A) Domain: all real numbers; Range: �1 � y � 1; Period: 2� (B) Domain: all real numbers; Range: �1 � y � 1; Period: 2�

(C) Domain: all real numbers except x � �/2 � k�, k an integerRange: all real numbers; Period: � (5-6)

43

54

35

�2 � 108

�2 � 108

2 � 108

2 � 108

�3�3�2�2�2�2�2�2�2�2

�2

�10

�15.2

10

15.2

P

�10

�15.2

10

15.2

5 100

�2

3�2

�

�3�3

x

y

w2

w1 � 2i

w3

�3

Answers A-133

49. 20x � 10y � 460 51. 10x � 30y � 28030x � 30y � 960 30x � 10y � 360

5x � 10y � 220 y � x � 0y � x � 0 x � y � 0x � y � 0

Exercise 8-5

1. Max z � 16 at (7, 9) 3. Max z � 84 at (0, 12) and (7, 9) (multiple optimal solutions) 5. Min z � 32 at (0, 8)7. Min z � 36 at (12, 0) and (4, 3) (multiple optimal solutions) 9. Max z � 18 at (4, 3) 11. Min z � 12 at (4, 0)

13. Max z � 52 at (4, 10) 15. Min z � 44 at (4, 4)17. Min z � 1,500 at (60, 0); max z � 3,000 at (60, 30) and (120, 0) (multiple optimal solutions)19. Min z � 300 at (0, 20); max z � 1,725 at (60, 15) 21. Max P � 5,507 at x � 6.62 and y � 4.2523. (A) a 2b (B) b � a � 2b (C) b 3a (D) a � 2b (E) b � 3a25. (A) 6 trick skis, 18 slalom skis; $780 (B) The maximum profit decreases to $720 when 18 trick and no slalom skis are produced.

(C) The maximum profit increases to $1,080 when no trick and 24 slalom skis are produced.27. 9 model A trucks, 6 model B trucks, $279,00029. (A) 40 tables, 40 chairs; $4,600 (B) The maximum profit decreases to $3,800 when 20 tables and 80 chairs are produced.31. (A) Max P � $450 when 750 gal is produced using the old process exclusively.

(B) The maximum profit decreases to $380 when 400 gal is produced using the old process and 700 gal using the new process.(C) The maximum profit decreases to $288 when 1,440 gal is produced using the new process exclusively.

33. The nitrogen will range from a minimum of 940 lb when 40 bags of brand A and 100 bags of brand B are used to a maximum of 1,190 lbwhen 140 bags of brand A and 50 bags of brand B are used.


1. (2, 3) (8-1) 2. No solution (inconsistent) (8-1) 3. Infinitely many solutions (t, (4t � 8)/3), for any real number t (8-1)4. (�1, 3), (5, �3) (8-3) 5. (1, �1), ( , � ) (8-3) 6. (1, 3), (1, �3), (�1, 3), (�1, �3) (8-3)7. (8-1) 8. (8-4) 9. (8-4)

10. (8-1) 11. (8-1) 12. (8-1) 13. x1 � 4 (8-2)x2 � �7(4, �7)

14. (8-2) 15. x1 � x2 � 4 (8-2) 16. Min z � 18 at (0, 6); max z � 42 at (6, 4) (8-5)x1 � t � 4, x2 � t, t any real number

No solution 0 � 1

x1 � x2 � 4

�1

0

�4

6 � 5

�3��1

1

�4

�2 �

5

4��3

1

�6

�4 � 12

5�

x

y

5

�5

�5 5

x

y

�5

105x

y

5

�5 5

(2, �1)

15

75

13

(10, 6)x

y

60

20

40

6020 40

30x � 10y � 360

10x � 30y � 280

(14, 18)

(20, 12) x

y

60

20

40

6020 40

20x � 10y � 460

5x � 10y � 220

30x � 30y � 960

A-134 Answers

17. x1 � 2, x2 � �2; each pair of lines has the same intersection point. (8-1)

x1 � x2 � 4 x1 � x2 � 4 x1 � x2 � 4 x1 � 22x1 � x2 � 2 x1 � 3x2 � �6 x1 � x2 � �2 x2 � �2

18. (2.54, 2.15) (8-1) 19. x1 � �1, x2 � 3 (8-2) 20. x1 � �1, x2 � 2, x3 � 1 (8-2) 21. x1 � 2, x2 � 1, x3 � �1 (8-2)22. Infinitely many solutions: x1 � �5t � 12, x2 � 3t � 7, x3 � t, t any real number (8-2) 23. No solution (8-2)24. Infinitely many solutions: x1 � � t � , x2 � t � , x3 � t, t any real number (8-2)25. (2, ), (2, � ), (�1, i), (�1, �i) (8-3) 26. (1, �2), (�1, 2), (2, �1), (�2, 1) (8-3)27. (2, �2), (�2, 2), ( , ), (� , � ) (8-3)28. Corner points: (0, 0), (0, 4), 29. Corner points: (0, 8), ( , ), 30. Corner points: (4, 4), (10, 10),

(3, 2), (4, 0) (8-4) (12, 0) (8-4) (20, 0) (8-4)Bounded Unbounded Bounded

31. Max z � 46 at (4, 2) (8-5) 32. Min z � 75 at (3, 6) and (15, 0) (multiple optimal solutions) (8-5)33. Min z � 44 at (4, 3); max z � 82 at (2, 9) (8-5) 34. x1 � 1,000, x2 � 4,000, x3 � 2,000 (8-2)35. (2, 2), (�2, �2), ( , � ), (� , ) (8-3) 36. Max z � 26,000 at (600, 400) (8-5)37. (�2.16, �0.37), (�1.09, 5.59), (1.09, �5.59), (2.16, 0.37) (8-3)38. (A) A unique solution (B) No solution (C) An infinite number of solutions (8-2)39. -lb packages: 48, -lb packages: 72 (8-1, 8-2) 40. 6 by 8 m (8-3) 41. 40 g mix A, 60 g mix B, 30 g mix C (8-2)42. (A) 22 nickels, 8 dimes (B) 22 � 3t nickels, 8 � 4t dimes, and t quarters, t � 0, 1, 2 (8-1, 8-2)43. (A) Maximum profit is P � $7,800 when 80 regular and 30 competition sails are produced.

(B) The maximum profit increases to $8,750 when 70 competition and no regular sails are produced.(C) The maximum profit decreases to $7,200 when no competition and 120 regular sails are produced. (8-5)

44. (A) The minimum cost is C � $13 when 100 g of mix A and 150 g of mix B are used.(B) The minimum cost decreases to $9 when 50 g of mix A and 275 g of mix B are used.(C) The minimum cost increases to $28.75 when 250 g of mix A and 75 g of mix B are used. (8-5)


1. 3. 5. Not defined 7. 9. 11. [41] 13.

15. 17. 19. [�36] 21. 23. [�13] 25. 27. Not defined

29. 31. Not defined 33. 35. 37. Not defined

39. 41. 43. 45. a � 4, b � 5, c � �9, d � �4

47. x � 5, y � 2 49. a � 1, b � �4, c � 0, d � 1 51. a � �3, b � 6, c � 2, d � �1 53. True 55. True

�32

42

28

�78��6

�189

�226

4

�62

�76

�12

�42

�38

��24

�45

�110

8

122

110

24

�198

�234

��

24

�14

�4

�32

18

56��

�19

15

25

�2

4

10

�14

6

�4�� 3

�31

2

�10

�6

�8�

�5

�10

30

0

0

0

�3

6

�18��6

15

12

�30�� 16

�10

�18

18��13

26

8

48�

� 3

�13�� 12

�8

�16

36

28

20��3

1

�5

1

6

8��3

�2

12

2

8

�5��2

4

5

�6�

13

12

27�74

7�727�74

7�7

x

y

10

10 20

x

y

5

10

5 10x

y

5

5

165

125

�2�2�2�2�2�2

97

57

47

37

5

�5

�5 5x

1

x2

(2, �2)

5

�5 5x

1

x2

(2, �2)

5

�5 5x

1

x2

(2, �2)

5

�5 5x

1

x2

(2, �2)

Answers A-137


1. (9-1) 2. [�11] (9-1) 3. [�15 19] (9-1) 4. (9-1) 5. (9-1)

6. Not defined (9-1) 7. Not defined (9-1) 8. (9-1) 9. [�15 18] (9-1) 10. (9-2)

11. (A) x1 � �1, x2 � 3 (B) x1 � 1, x2 � 2 (C) x1 � 8, x2 � �10 (9-3) 12. �17 (9-4) 13. 0 (9-4, 9-5)

14. x � 2, y � �1 (9-6) 15. (A) �2 (B) 6 (C) 2 (9-5) 16. (9-1)

17. (9-1) 18. (9-1) 19. [16] (9-1) 20. Not defined (9-1)

21. (9-1) 22. (9-2)

23. (A) x1 � 2, x2 � 1, x3 � �1 (B) x1 � 1, x2 � �2, x3 � 1 (C) x1 � �1, x2 � 2, x3 � �2 (9-3)24. � (9-4) 25. 35 (9-4, 9-5) 26. y � � 2 (9-6)27. (A) A unique solution (B) Either no solution or an infinite number (9-3)

28. No. (9-3) 29. X � (A � C )�1B (9-3) 30. or (9-2)

31. x1 � 1,000, x2 � 4,000, x3 � 2,000 (9-3) 32. 42 (9-5)

33. � (u � kv)x � (w � kx)v � ux � kvx � wv � kvx � ux � wv � (9-5)

34. Theorem 4 in Section 9-5 implies that both points satisfy the equation. All other points on the line through the given points will alsosatisfy the equation. (9-5)

35. (A) 60 tons at Big Bend, 20 tons at Saw Pit (B) 30 tons at Big Bend, 50 tons at Saw Pit(C) 40 tons at Big Bend, 40 tons at Saw Pit (9-3)

36. (A) $27 (B) Elements in LH give the total cost of manufacturing each product at each plant.(C) (9-1)

37. (A) (B) (C)(9-1)

38. GRAPHING UTILITY (9-2)


1. x � 2, y � �1 (8-1) 2. (�1, 2) (8-1) 3. (� , � ), (1, 1) (8-3) 4. (8-4)

5. Maximum: 33; Minimum: 10 (8-5)

6. (A) (B) Not defined (C) [3] (D) (E) [�1, 8] (F) Not defined (9-1)

7. �10 (9-4) 8. (A) x1 � 3, x2 � �4 (B) x1 � 2t � 3, x2 � t, t any real number. (C) No solution (8-1)

9. (A) (B) (C) x1 � �1, x2 � 4 (8-1, 8-2)

10. (A) (B) (C) x1 � 13, x2 � 5 (D) x1 � �11, x2 � �4 (9-3)A�1 � ��5

�2

3

1��1

2

�3

�5��x1

x2� � �k1

k2�

�1

0

0

1 � �1

4�� 1

�1

1

1 � 3

5�

�1

4

7

�7��0

3

�3

�9�

x

y

5

5

75

15

�3,150

1,550Total production of each item in January

Desks

Stands��200

80

160

40��1,600

890

1,730

720�

�$46.35

$30.45LH �

NorthCarolina

$41.00

$27.00

SouthCarolina

Desks

Stands�

�u

w

v

x�� u � kv

w � kx

v

x�

1

12 ��11

10

1

�1

2

�1

60

�48

0��

�11121012112

� 112212

� 112

5

�4

0�

105

1112

��1

�212

1

3

�14

1

2

�14�� 63

�42

�24

16

�39

26�

�12

0

�8

24

0

�16

�6

0

4��22

38

19

42�

�7

28

�21

16

40

�8

�9

�30

17�

� 2

�1

7

�4��13

20

�29

�24�� 3

�4

3

9�� 16

�6�� 4

�12

8

18�

A-140 Answers


1. (A) Geometric (B) Arithmetic (C) Arithmetic (D) Neither (E) Geometric (10-1, 10-3)2. (A) 5, 7, 9, 11 (B) a10 � 23 (C) S10 � 140 (10-1, 10-3)3. (A) 16, 8, 4, 2 (B) a10 � (C) S10 � 31 (10-1, 10-3)4. (A) �8, �5, �2, 1 (B) a10 � 19 (C) S10 � 55 (10-1, 10-3)5. (A) �1, 2, �4, 8 (B) a10 � 512 (C) S10 � 341 (10-1, 10-3) 6. S � 32 (10-3) 7. 3,628,800 (10-4)8. 17,100,720 (10-4) 9. 1,287 (10-4) 10. P8,4 � 1,680; C8,4 � 70 (10-5)

11. (A) 12 combined outcomes: (B) 6 � 2 � 12 (10-5) 12. 6 � 5 � 4 � 3 � 2 � 1 � 720 (10-5) 13. P6,6 � 6! � 720 (10-5)

14. P1: 5 � 12 � 4 � 1; P2: 5 � 7 � 22 � 4 � 2; P3: 5 � 7 � 9 � 32 � 4 � 3 (10-2)15. P1: 2 � 21�1 � 2; P2: 2 � 4 � 22�1 � 2; P3: 2 � 4 � 8 � 23�1 � 2 (10-2)16. P1: 491 � 1 � 48 is divisible by 6; P2: 492 � 1 � 2,400 is divisible by 6; P3: 493 � 1 � 117,648 is divisible by 6 (10-2)17. Pk: 5 � 7 � 9 � . . . � (2k � 3) � k2 � 4k; Pk�1: 5 � 7 � 9 � . . . � (2k � 3) � (2k � 5) � (k � 1)2 � 4(k � 1) (10-2)18. Pk: 2 � 4 � 8 � . . . � 2k � 2k�1 � 2; Pk�1: 2 � 4 � 8 � . . . � 2k � 2k�1 � 2k�2 � 2 (10-2)19. Pk: 49k � 1 � 6r for some integer r; Pk�1: 49k�1 � 1 � 6s for some integer s (10-2) 20. n � 31 is a counterexample (10-2)

21. S10 � �6 � 4 � 2 � 0 � 2 � 4 � 6 � 8 � 10 � 12 � 30 (10-3) 22. S7 � 8 � 4 � 2 � 1 � � � � 15 (10-3)

23. S � (10-3) 24. ; S � (10-3) 25. C6,3 � 20 (10-5) 26. d � 3, a5 � 25 (10-3)

27. a1 � �9, d � (10-3) 28. 336; 512; 392 (10-5) 29. (10-3) 30. (A) P6,3 � 120 (B) C5,2 � 10 (10-5)

31. 190 (10-4) 32. 1,820 (10-4) 33. 1 (10-4) 34. (10-4) 35. (10-5)

36. x5 � 5x4y � 10x3y2 � 10x2y3 � 5xy4 � y5 (10-4) 37. �1,760x3y9 (10-4) 41. 29 (10-4) 42. 26 (10-1)

43. 25 � 32; 6 (10-5) 44. 49 g/2 ft; 625 g/2 ft (10-3) 45. 12 (10-5)

46. x6 � 6ix5 � 15x4 � 20ix3 � 15x2 � 6ix � 1 (10-4) 47. P5,5 � 120 (10-5) 53. n � 5 (10-1)

54. {an} is neither arithmetic nor geometric; {bn} is geometric (10-3)


1. 3. 5. 7.

x

y

5

�5

�5 5

Directrixy � 1F(0, �1)

x

y

5

�5 5

Directrixx � 3

F(�3, 0)x

y

5

�5

�5 5

Directrixy � �2

F(0, 2)

x

y

5

�5 5

Directrixx � �1

F(1, 0)

�1000

501 �987!493!

811

195

14Sn � �

n

k�1

(�1)k�1

3k815

78

18

14

12

HT

HT

HT

HT

HT

HT

1

2

3

4

5

6

3132

132

A-144 Answers

23. y2 � �8(x � 1), �1 � x � 1; 25. x2 � y2 � 2x, x � 0 or (x � 1)2 � y2 � 1, x � 0; 27.part of a parabola circle (note hole at origin)

29. ; ellipse with center (3, 2) 31. ; hyperbola with center (�3, �1) 33. 5x � 2y � 0

35. ; hyperbola with center (4, �1); 37. ; ellipse with center (3, �4);

x � 4 � 3 tan t, y � �1 � 5 sec t, � � t � , t �x � 3 � 7 cos t, y � �4 � 2 sin t, 0 � t � 2�

39. (A) (4.8, �1.5) 41. (A) 43.292 s (B) 9,183.619 m; 9.184 km (C) 2,295.918 m(B) x2 � y2 � 25; circle


1. Foci: F�(�4, 0), F(4, 0) (11-2) 2. (11-1) 3. Foci: F�(0, � ), F(0, ) (11-3)Major axis length � 10 Transverse axis length � 6Minor axis length � 6 Conjugate axis length � 10

4. (A) (B) Hyperbola (11-4) 5. (A) (x � 5)2 � �12(y � 4) (B) Parabola (11-4)

6. (A) (B) Ellipse (11-4) 7. y2 � �7x (11-1) 8. (11-2) 9. (11-3)x2

25 �

y2

11 � 1

x2

4 �

y2

965 � 1

(x � 6)2

9 �

(y � 4)2

16 � 1

(y � 2)2

25 �

(x � 4)2

4 � 1

x

y

10

�10

�10 10

F

F�

x

y

F(0, �3)

Directrixy � 3

5

�5

�5 5

x

y

5

�5

�5 5

FF�

�34�34

x

y5

�5

�5 5

�10

�10

10

10

�10

�10

10

10

�

2

3�

2

�

2

(x � 3)2

49 �

(y � 4)2

4 � 1

(y � 1)2

25 �

(x � 4)2

9 � 1

�10

�10

10

10

�10

�10

10

10

(y � 1)2

25 �

(x � 3)2

4 � 1

(x � 3)2

36 �

(y � 2)2

16 � 1

x

y

2

�2

�2 2x

y

5

�5

�5 5

x

y

1

2

2��3

2�3

4�3

5�3

Answers A-145

10. y � x � 1, x � 0; a ray (part of a straight line) (11-5) 11. (11-2, 8-3)

12. (11-1, 11-2, 8-3) 13. (11-2, 8-3)

14. ; 15. (x � 2)2 � 8(y � 3); 16. ;

ellipse (11-4)parabola (11-4)

hyperbola (11-4)

17. ; ellipse (11-5) 18. m � 0.2; x2 � 50y is a magnification by a factor 50 of x2 � y (11-1)

19. (y � 4)2 � �8(x � 4), or y2 � 8y � 8x � 16 � 0 (11-1) 20. ; hyperbola (11-3)

21. ; ellipse (11-2) 22. F�(�3, � � 2), F(�3, � 2) (11-4) 23. F(2, �1) (11-4)

24. F�(� � 3, �2), F( � 3, �2) (11-4)

25. y � , x 0; hyperbola (one branch) (11-5)

26. (2.09, 2.50), (3.67, �1.92) (11-4) 27. 4 ft (11-1) 28. (11-2) 29. 4.72 ft deep (11-3)x2

52 � y2

32 � 1

x

y

5

�5

�5 5

1

x

�13�13

�12�12x2

36 �

y2

20 � 1

x2

4 �

y2

12 � 1

x

y

5

�5 5

(x � 2)2

4 �

(y � 3)2

16 � 1

x �

y � y

�5

�10 5x

0�

x �0�

y �

x

y

5

�5

�5 105x �0�

y �

x

y

5

�5

(x � 3)2

9 �

(y � 2)2

4 � 1

(x � 3)2

4 �

(y � 2)2

16 � 1

x

y

5

�5

�5 5

(�1, �3) (1, �3)

(�1, 3) (1, 3)

x

y

5

�5

�5 5

(�3, 3.2) (3, 3.2)

x

y

5

�5

�5 5

x

y

5

�5(4, �2)

(�4, 2)

12

Answers A-149

67. 69. 0.1816 71. 3.557 73. 0.092 75 75. 1.602 77. Both are 1.259 79. Both are 0.4807

81. x � 0 83. All real numbers 85. A and E, B and F, C and D 87.

89. 91.

93.

Appendix A Review Exercise

1. (A) T (B) T (C) F (D) T (E) F (F) F (A-1) 2. (A) (y � z)x (B) (2 � x) � y (C) 2x � 3x (A-1)3. x3 � 3x2 � 5x � 2 (A-2) 4. x3 � 3x2 � 3x � 22 (A-2) 5. 3x5 � x4 � 8x3 � 24x2 � 8x � 64 (A-2) 6. 3 (A-2)7. 1 (A-2) 8. 14x2 � 30x (A-2) 9. 9m2 � 25n2 (A-2) 10. 6x2 � 5xy � 4y2 (A-2) 11. 4a2 � 12ab � 9b2 (A-2)

12. (3x � 2)2 (A-3) 13. Prime (A-3) 14. 3n(2n � 5)(n � 1) (A-3) 15. (12a3b � 40b2 � 5a)/(30a3b2) (A-4)16. (7x � 4)/[6x(x � 4)] (A-4) 17. ( y � 2)/[ y( y � 2)] (A-4) 18. u (A-4) 19. 6x5y15 (A-5) 20. 3u4/v2 (A-5)21. 6 � 102 (A-5) 22. x6/y4 (A-5) 23. u7/3 (A-6) 24. 3a2/b (A-6) 25. (A-7) 26. �3(xy)2/3 (A-7)27. (A-7) 28. (A-7) 29. (A-7) 30. (A-7) 31. (A-7)32. {�3, �1, 1} (A-1) 33. Subtraction (A-1) 34. Commutative (�) (A-1) 35. Distributive (A-1)36. Associative (�) (A-1) 37. Negatives (A-1) 38. Identity (�) (A-1) 39. (A) T (B) F (A-1)40. 0 and �3 are two examples of infinitely many. (A-1) 41. (A) (a) and (d) (B) None (A-2) 42. 4xy � 2y2 (A-2)43. m4 � 6m2n2 � n4 (A-2) 44. 10xh � 5h2 � 7h (A-2) 45. 2x3 � 4x2 � 12x (A-2) 46. x3 � 6x2y � 12xy2 � 8y3 (A-2)47. (x � y)(7x � y) (A-3) 48. Prime (A-3) 49. 3xy(2x2 � 4xy � 5y2) (A-3) 50. ( y � b)( y � b � 1) (A-3)

51. 3(x � 2y)(x2 � 2xy � 4y2) (A-3) 52. ( y � 2)( y � 2)2 (A-3) 53. x(x � 4)2(5x � 8) (A-3) 54. (A-4)

55. 2m/[(m � 2)(m � 2)2] (A-4) 56. y2/x (A-4) 57. (x � y)/(x � y) (A-4) 58. �ab/(a2 � ab � b2) (A-4)59. Incorrect; correct final form is (x2 � 2x � 2)/(x � 1) (A-4) 60. (A-5) 61. (A-5) 62. 3x2/(2y2) (A-6)63. 27a1/6/b1/2 (A-6) 64. x � 2x1/2y1/2 � y (A-6) 65. 6x � 7x1/2y1/2 � 3y (A-6) 66. 2 � 10�7 (A-5)67. 3.213 � 106 (A-5) 68. 4.434 � 10�5 (A-5) 69. �4.541 � 10�6 (A-5) 70. 128,800 (A-6) 71. 0.01507 (A-6)72. 0.3664 (A-7) 73. 1.640 (A-7) 74. 0.08726 (A-6) 75. (A-7) 76. (A-7)

77. (A-7) 78. (A-7) 79. (A-7) 80. (A-7)

81. (A-7) 82. (A-7) 83. (A-7) 84. (A-7)85. 2 � x�1/2 (A-7) 86. ; rational (A-1) 87. (A) {�4, �3, 0, 2} (B) {�3, 2} (A-1) 88. 0 (A-7)89. x3 � 8x2 � 6x � 1 (A-2) 90. x(2a � 3x � 4)(2a � 3x � 4) (A-3) 91. All three have the same value. (A-7)92. (x � 2)(x � 3)4 (A-5) 93. a2b2/(a3 � b3) (A-5) 94. x � y (A-6) 95. xm�1 (A-6)

96. (A-7) 97. (A-7) 98. xn�1 (A-7)99. Volume � 3�(x � 2)2 � 3�x2 � 12�x � 12� ft3 (A-2) 100. 2.84 � 104 dollars per person; $28,400 per person (A-5)

101. (A) 24,000 units (B) Production doubles to 48,000 units(C) At any production level, doubling the units of capital and labor doubles production. (A-6)

102. (A-4)

103. (A) A � 480 � 6x2 � 6(80 � x2) (B) V � x(16 � 2x)(15 � 1.5x) � 240x � 54x2 � 3x3 (A-3)

R � R1R2R3

R2R3 � R1R3 � R1R2

1/(�3 t2 � �3 5t � �3 25)(1 � �3 x � �3 x2)/(1 � x)

23

611

32

1/(�t � �5)�y2 � 4 � 2(4u � 12�uv � 9v)/(4u � 9v)(6x � 3�xy)/(4x � y)

2x � 3�xy � 5y�3 2x2y�3 2x2y�5 12x3y2/(2x)

x�3 2x2�6x2y2�5 3x2y

59

14

(x � 4)(x � 2)2

x3

�4 y3(3�5 � 5)/42b�3a6x2y3�xy3x2y�3 x2y3�5 x2

�kn

xkm � (xkm)1/kn � xkm/kn � x m/n � �n

x m

1

�3 (x � h)2 � �3 x(x � h) � �3 x2

(�x � �y � �z)[(x � y � z) � 2�xy]

(x � y � z)2 � 4xy

�3 a2 � �3 ab � �3 b2

a � b

1/(�x � h � �x)

exercise 1-8 - mcgraw hill education€¦ · 5 12 01 1 2 1 5 5 2 2 2 1 2 1 5 5 2 2 2 x 4 0 2 [x...

Documents