Solution manual-for-trigonometry-9th-edition-by-larson

2 © 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

C H A P T E R P Prerequisites

Section P.1 Review of Real Numbers and Their Properties

1. irrational

2. origin

3. absolute value

4. composite

5. terms

6. Zero-Factor Property

7. 7 22 3

9, , 5, , 2, 0, 1, 4, 2, 11

(a) Natural numbers: 5, 1, 2

(b) Whole numbers: 0, 5, 1, 2

(c) Integers: 9, 5, 0, 1, 4, 2, 11

(d) Rational numbers: 7 22 3

9, , 5, , 0, 1, 4, 2, 11

(e) Irrational numbers: 2

8. 7 53 4

5, 7, , 0, 3.12, , 3, 12, 5

(a) Natural numbers: 12, 5

(b) Whole numbers: 0, 12, 5

(c) Integers: 7, 0, 3, 12, 5

(d) Rational numbers: 7 53 4

7, , 0, 3.12, , 3, 12, 5


9. 2.01, 0.666 . . ., 13, 0.010110111 . . ., 1, 6

(a) Natural numbers: 1

(b) Whole numbers: 1

(c) Integers: 13, 1, 6

(d) Rational numbers: 2.01, 0.666 . . ., 13, 1, 6

(e) Irrational numbers: 0.010110111 . . .

10. 12 15 2

25, 17, , 9, 3.12, , 7, 11.1, 13

(a) Natural numbers: 25, 9, 7, 13

(b) Whole numbers: 25, 9, 7, 13

(c) Integers: 25, 17, 9, 7, 13

(d) Rational numbers:

125

25, 17, , 9, 3.12, 7, 11.1, 13


11. (a)

(b)

(c)

(d)

12. (a)

(b)

(c)

(d)

13. 4 8

14. 163

1

15. 5 26 3

16. 8 37 7

17. (a) The inequality 5x denotes the set of all real numbers less than or equal to 5.

(b)

(c) The interval is unbounded.

NOT FOR SALE

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Section P.1 Review of Real Numbers and Their Properties 3

© 2014 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

18. (a) The inequality 0x denotes the set of all real numbers less than zero.

(b)


19. (a) The interval 4, denotes the set of all real

numbers greater than or equal to 4.

(b)


20. (a) , 2 denotes the set of all real numbers less

than 2.

(b)


21. (a) The inequality 2 2x denotes the set of all real numbers greater than 2 and less than 2.

(b)

(c) The interval is bounded.

22. (a) The inequality 0 6x denotes the set of all real numbers greater than zero and less than or equal to 6.

(b)


23. (a) The interval 5, 2 denotes the set of all real

numbers greater than or equal to 5 and less than 2.

(b)


24. (a) The interval 1, 2 denotes the set of all real

numbers greater than 1 and less than or equal to 2.

(b)


25. 0; 0,y

26. 25; , 25y

27. 10 22; 10, 22t

28. 3 5; 3, 5k

29. 65; 65,W

30. 2.5% 5%; 2.5%, 5%r

31. 10 10 10

32. 0 0

33. 3 8 5 5 5

34. 4 1 3 3

35. 1 2 1 2 1

36. 3 3 3 3 6

37. 5 5 5

15 5 5

38. 3 3 3 3 9

39. If 2,x then 2x is negative.

So, 2 2

1.2 2

x x

x x

40. If 1,x then 1x is positive.

So, 1 1

1.1 1

x x

x x

41. 4 4 because 4 4 and 4 4.

42. 5 5 since 5 5.

43. 6 6 because 6 6 and

6 6 6.

44. 2 2 because 2 2.

45. 126, 75 75 126 51d

46. 126, 75 75 126

51

d

47. 5 5 52 2 2, 0 0d

48. 51 11 11 14 4 4 4 2,d

49. 16 16 128112 1125 75 75 5 75

,d

50. 9.34, 5.65 5.65 9.34 14.99d

51. , 5 5d x x and , 5 3,d x so 5 3.x

NOT FOR SALESection P.1 Review of Real Numbers and Theiection P.1 Review of Real Numbers and The



4 Chapter P Prerequisites


52. , 10 10 ,d x x and , 10 6,d x so

10 6.x

53. ,d y a y a and , 2,d y a so 2.y a

54. 60 23 37 F

Receipts, R Expenditures, E R E

55. $1880.1 $2292.8 1880.1 2292.8 $412.7 billion

56. $2406.9 $2655.1 2406.9 2655.1 $248.2 billion

57. $2524.0 $2982.5 2524.0 2982.5 $458.5 billion

58. $2162.7 $3456.2 2162.7 3456.2 $1293.5 billion

59. 7 4x

Terms: 7 , 4x

Coefficient: 7

60. 36 5x x

Terms: 36 , 5x x

Coefficient: 6, 5

61. 34 52

xx

Terms: 34 , , 52

xx

Coefficients: 1

4,2

62. 23 3 1x

Terms: 23 3 , 1x

Coefficients: 3 3

63. 4 6x

(a) 4 1 6 4 6 10

(b) 4 0 6 0 6 6

64. 9 7x

(a) 9 7 3 9 21 30

(b) 9 7 3 9 21 12

65. 2 5 4x x

(a) 2

1 5 1 4 1 5 4 10

(b) 2

1 5 1 4 1 5 4 0

66. 1

1

x

x

(a) 1 1 2

1 1 0

Division by zero is undefined.

(b) 1 1 0

01 1 2

67. 1

6 1, 66

h hh

Multiplicative Inverse Property

68. 3 3 0x x

Additive Inverse Property

69. 2 3 2 2 3x x

Distributive Property

70. 2 0 2z z

Additive Identity Property

71. 3 3 Associative Property of Multiplication

3 Commutative Property of Multiplication

x y x y

x y

72. 1 17 7

7 12 7 12 Associative Property of Multiplication

1 12 Multiplicative Inverse Property

12 Multiplicative Identity Property

NOT FOR SALErequisitesrequisites



Section P.2 Solving Equations 5


73. 5 5 15 10 9 31 48 12 6 24 24 24 24 8

74. 4 18 2

6 6 3

75. 2 8 3 5

3 4 12 12 12

x x x x x

76. 5 2 5 1 5

6 9 3 9 27

x x x

77. (a) Because 0, 0.A A

The expression is negative.

(b) Because , 0.B A B A

The expression is negative.

(c) Because 0, 0.C C

The expression is positive.

(d) Because , 0.A C A C

The expression is positive.

78. The types of real numbers that may be used in a range of prices are nonnegative rational numbers because a price is neither negative nor irrational. For example, a price

of 1.80 or $ 2 would not be reasonable. The types of real numbers to describe a range of lengths are nonnegative and rational because lengths are often not in whole unit amounts but in parts, such as 2.3 centimeters

or 3

4inch.

79. False. Because 0 is nonnegative but not positive, not every nonnegative number is positive.

80. False. Two numbers with different signs will always have a product less than zero.

81. (a)

(b) (i) As n approaches 0, the value of 5 n increases

without bound (approaches infinity).

(ii) As n increases without bound (approaches infinity), the value of 5 n approaches 0.

Section P.2 Solving Equations

1. equation

2. 0ax b

3. extraneous

4. factoring; square roots; completing; square; Quadratic Formula

5. 11 15

11 11 15 11

4

x

x

x

6. 7 19

7 19

7 19

7 19 19 19

12

x

x x x

x

x

x

7. 7 2 25

7 7 2 25 7

2 18

2 182 2

9

x

x

x

x

x

8. 3 5 2 7

3 2 5 2 2 7

5 7

5 5 7 5

12

x x

x x x x

x

x

x

9. 4 2 5 7 6

4 5 2 7 6

2 7 6

6 2 7 6 6

5 2 7

5 2 2 7 2

5 5

5 5

5 51

y y y

y y y

y y

y y y y

y

y

y

y

y

10. 0.25 0.75 10 3

0.25 7.5 0.75 3

0.50 7.5 3

0.50 4.5

9

x x

x x

x

x

x

n 0.0001 0.01 1 100 10,000

5 n 50,000 500 5 0.05 0.0005

NOT FOR SALESection P.2 SolvinSection P.2 Solvi





11. 3 2 3 8 5

6 9 8 5

5 9 8 5

5 5 9 8 5 5

9 8

x x x

x x x

x x

x x x x

No solution

12. 9 10 5 2 2 5

9 10 5 4 10

9 10 9 10

x x x

x x x

x x

The solution is the set of all real numbers.

13. 3 4

48 3

9 324

24 2423

424

23 24 244

24 23 23

96

23

x x

x x

x

x

x

or 3 4

48 3

3 424 24 4

8 3

9 32 96

23 96

96

23

x x

x x

x x

x

x

The second method is easier. The fractions are eliminated in the first step.

14. 5 1 14 2 2

5 1 14 4

4 2 2

5 1 14 4 4 4

4 2 2

5 2 4 2

4

xx

xx

xx

x x

x

15. 5 4 25 4 3

3 5 4 2 5 4

15 12 10 8

5 20

4

xx

x x

x x

x

x

16. 10 3 1

5 6 2

2 10 3 1 5 6

20 6 5 6

15 0

0

x

x

x x

x x

x

x

17. 13 5

10 4

10 13 4 5

10 13 4 5

6 18

3

x xx x

x xx x

x

x

18.1 2

0 Multiply both sides by 5 .5

1 5 2 0

3 5 0

3 5

53

x xx x

x x

x

x

x

19.4

2 04 4

42 0

41 2 0

3 0

xx x

xx

Contradiction; no solution

20.

2 2

7 84 Multiply both sides by 2 1 2 1 .

2 1 2 17 2 1 8 2 1 4 2 1 2 1

14 7 16 8 16 4

6 11

11

6

xx x

x xx x x x x

x x x x

x

x






21.2 1 2

Multiply both sides by 4 2 .4 2 4 2

2 1 2 2 4

2 2 2 8

2 3 10

12 3

4

x xx x x x

x x

x x

x

x

x

A check reveals that 4x is an extraneous solution—it makes the denominator zero. There is no real solution.

22.4 6 15

Multiply both sides by 1 3 1 .1 3 1 3 1

4 6 151 3 1 1 3 1 1 3 1

1 3 1 3 14 3 1 6 1 15 1

12 4 6 6 15 15

18 2 15 15

3 13

13

3

x xx x x

x x x x x xx x x

x x x

x x x

x x

x

x

23.2

1 1 103 3 9

1 1 10Multiply both sides by 3 3 .

3 3 3 3

1 3 1 3 10

2 10

5

x x x

x xx x x x

x x

x

x

24. 2

1 3 42 3 6

1 3 4Multiply both sides by 3 2 .

2 3 3 2

3 3 2 4

3 3 6 4

4 3 4

4 7

7

4

x x x x

x xx x x x

x x

x x

x

x

x

25. 26 3 0

3 2 1 0

x x

x x

12

3 0 or 2 1 0

0 or

x x

x x

26. 29 1 0

3 1 3 1 0

x

x x

13

13

3 1 0

3 1 0

x x

x x

27. 2 2 8 0

4 2 0

x x

x x

4 0 or 2 0

4 or 2

x x

x x

28. 2 10 9 0

9 1 0

x x

x x

9 0 9

1 0 1

x x

x x


INSTRUCTOR USE ONLY 33

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29. 2 10 25 0

5 5 0

5 0

5

x x

x x

x

x

30. 24 12 9 0

2 3 2 3 0

x x

x x

322 3 0x x

31. 2

2

4 12

4 12 0

6 2 0

x x

x x

x x

6 0 or 2 0

6 or 2

x x

x x

32. 2

2

2

2

8 12

8 12 0

1 8 12 1 0

8 12 0

6 2 0

x x

x x

x x

x x

x x

6 0 6

2 0 2

x x

x x

33. 2

2

2

34

34

8 20 0

4 8 20 4 0

3 32 80 0

3 20 4 0

x x

x x

x x

x x

203

3 20 0 or 4 0

or 4

x x

x x

34. 2

2

18 16 0

8 128 0

16 8 0

x x

x x

x x

16 0 16

8 0 8

x x

x x

35. 2 49

7

x

x

36. 2 32

32 4 2

x

x

37. 2

2

3 81

27

3 3

x

x

x

38. 2

2

9 36

4

4 2

x

x

x

39. 2

12 16

12 4

12 4

16 or 8

x

x

x

x x

40. 2

9 24

9 24

9 2 6

x

x

x

41. 2

2 1 18

2 1 18

2 1 3 2

1 3 22

x

x

x

x

42. 2 2

7 3

7 3

x x

x x

7 3 or 7 3

7 3 or 2 4

2

x x x x

x

x

The only solution of the equation is 2.x

43. 2

2

2 2 2

2

4 32 0

4 32

4 2 32 2

2 36

2 6

2 6

4 or 8

x x

x x

x x

x

x

x

x x

44. 2

2

2 22

2

2 3 0

2 3

2 1 3 1

1 4

1 4

1 2

3 or 1

x x

x x

x x

x

x

x

x x


INSTRUCTOR USE ONLY 3 or 13 or xx 3 or 3 or





45. 2

2

2 2 2

2

6 2 0

6 2

6 3 2 3

3 7

3 7

3 7

x x

x x

x x

x

x

x

46. 2

2

2 2

2

8 14 0

8 14

8 4 14 16

4 2

4 2

4 2

x x

x x

x x

x

x

x

47. 2

2

2 2 2

2

9 18 3

12

31

2 1 13

21

3

21

3

21

3

61

3

x x

x x

x x

x

x

x

x

48. 2

2

2

2

2 22

2

7 2 0

2 7 0

2 7 0

2 7

2 1 7 1

1 8

1 2 2

1 2 2

x x

x x

x x

x x

x x

x

x

x

49. 2

2

2

2 22

2

2 5 8 0

2 5 8

54

2

5 5 54

2 4 4

5 894 16

5 894 4

5 89

4 4

5 894

x x

x x

x x

x x

x

x

x

x

50. 2

2

2

2 22

2

3 4 7 0

3 4 7

4 7

3 3

4 2 7 2

3 3 3 3

2 25

3 9

2 5

3 32 5

3 37

1 or 3

x x

x x

x x

x x

x

x

x

x x

51. 22 1 0x x

2

2

42

1 1 4 2 1

2 2

1 3 1, 1

4 2

b b acx

a

52. 22 1 0x x

2

2

42

1 1 4 2 1

2 2

1 1 8

41 3 1

1,4 2

b b acx

a






53. 2

2

2 2 0

2 2 0

x x

x x

2

2

42

2 2 4 1 2

2 1

2 2 31 3

2

b b acx

a

54. 2 10 22 0x x

2

2

42

10 10 4 1 22

2 1

10 100 882

10 2 35 3

2

b b acx

a

55. 22 3 4 0x x

2

2

42

3 3 4 2 4

2 2

3 41 3 414 4 4

b b acx

a

56. 2

2

3 1 0

3 1 0

x x

x x

2

2

42

3 3 4 1 1

2 1

3 13 3 13

2 2 2

b b acx

a

57. 2

2

12 9 3

9 12 3 0

x x

x x

2

2

42

12 12 4 9 3

2 9

12 6 7 2 7

18 3 3

b b acx

a

58. 2

2

9 37 6

9 6 37 0

x x

x x

2

2

42

6 6 4 9 37

2 9

6 6 38 1 3818 3 3

b b acx

a

59. 29 30 25 0x x

2

2

42

30 30 4 9 25

2 9

30 0 5

18 3

b b acx

a

60. 2

2

28 49 4

49 28 4 0

x x

x x

2

2

42

28 28 4 49 4

2 49

28 0 2

98 7

b b acx

a

61. 2

2

8 5 2

2 8 5 0

t t

t t

2

2

42

8 8 4 2 5

2 2

8 2 6 62

4 2

b b act

a

62. 225 80 61 0h h

2

2

42

80 80 4 25 61

2 25

80 6400 6100

50

8 10 35 50

8 35 5

b b ach

a


INSTRUCTOR USE ONLY 5 55 5





63.2

2

5 2

12 25 0

y y

y y

2

2

42

12 12 4 1 25

2 1

12 2 116 11

2

b b acy

a

64. 2

2

2

6 2

12 36 2

14 36 0

z z

z z z

z z

2

2

42

14 14 4 1 36

2 1

14 527 13

2

b b acz

a

65. 25.1 1.7 3.2 0x x

2

1.7 1.7 4 5.1 3.2

2 5.1

0.976, 0.643

x

66. 20.005 0.101 0.193 0x x

2

2

42

0.101 0.101 4 0.005 0.193

2 0.005

0.101 0.006341

0.01

2.137, 18.063

b b acx

a

67. 2422 506 347 0x x

2

506 506 4 422 347

2 422

1.687, 0.488

x

68. 23.22 0.08 28.651 0x x

2

2

4

2

0.08 0.08 4 3.22 28.651

2 3.22

0.08 369.0312.995, 2.971

6.44

b b acx

a

69. 2

2

2 2 2

2

2 1 0 Complete the square.

2 1

2 1 1 1

1 2

1 2

1 2

x x

x x

x x

x

x

x

70. 2

2

11 33 0 Factor.

11 3 0

3 0

x x

x x

x x

0 or 3 0

3

x x

x

71. 2

3 81 Extract square roots.

3 9

x

x

3 9 or 3 9

6 or 12

x x

x x

72. 2

14 49 0 Extract square roots.

7 0

7 0

7

x x

x

x

x

73. 2

2

2 22

2

114

114

1 11 12 4 2

1 122 4

1 122 4

12

0 Complete the square.

3

x x

x x

x x

x

x

x






74. 2

22

2

34

3 3 92 4 4

32

32

32

3 0 Complete the square.

3

3

3

3

x x

x x

x

x

x

75. 2 2

22

12

1 Extract square roots.

1

1

For 1 :

0 1 No solution

For 1 :

2 1

x x

x x

x x

x x

x x

x

x

76. 2

2

2

3 4 2 7 Quadratic Formula

0 2 3 11

3 3 4 2 11

2 2

3 974

3 974 4

x x

x x

x

77. 4 2

2 2

2

2

6 14 0

2 3 7 0

2 0 0

213 7 0

3

x x

x x

x x

x x

78. 3

2

53

53

36 100 0

4 9 25 0

4 3 5 3 5 0

4 0 0

3 5 0

3 5 0

x x

x x

x x x

x x

x x

x x

79. 3 2

2

2

5 3 45 0

5 6 9 0

5 3 0

5 0 0

3 0 3

x x x

x x x

x x

x x

x x

80. 3 2

2

2

3 3 0

3 3 0

3 1 0

3 3 1 0

3 0 3

1 0 1

1 0 1

x x x

x x x

x x

x x x

x x

x x

x x

81. 3 12 0

3 12

3 144

48

x

x

x

x

82. 10 4 0

10 0

10 16

26

x

x

x

x

83. 3

3

2 5 3 0

2 5 3

2 5 27

2 32

16

x

x

x

x

x

84. 3

3

1243

3 1 5 0

3 1 5

3 1 125

3 124

x

x

x

x

x

85.

2

2

26 11 4

4 26 11

16 8 26 11

3 10 0

5 2 0

5 0 5

2 0 2

x x

x x

x x x

x x

x x

x x

x x

86.

2

2

31 9 5

31 9 5

31 9 25 10

0 6

0 3 2

0 3 3

2 2

x x

x x

x x x

x x

x x

x x

x x x






87.

2 2

5 1

1 5

1 5

1 2 5 5

4 2 5

2 5

4 5

9

x x

x x

x x

x x x

x

x

x

x

88.

2 2

2

2

2 1 2 3 1

2 1 1 2 3

2 1 1 2 3

4 1 1 2 2 3 2 3

2 2 2 3

2 3

2 3

2 3 0

3 1 0

3 0 3

1 0 1, extraneous

x x

x x

x x

x x x

x x

x x

x x

x x

x x

x x

x x

89. 3 2

3 2

3

5 8

5 8

5 64

5 4 9

x

x

x

x

90. 2 3

2 3

2 9

2 9

2 729

2 27 29, 25

x

x

x

x

91. 3 22

32 2

32 2

2

2

5 27

5 27

5 27

5 9

14

14

x

x

x

x

x

x

92. 3 22

2 2 3

2

2

2

22 27

22 27

22 9

31 0

1 1 4 1 31 1 125 1 5 5

2 1 2 2

x x

x x

x x

x x

x

93. 2 5 11

2 5 11 8

2 5 11 3

x

x x

x x

94.

53

3 2 7

3 2 7

3 2 7

3 2 7 3

x

x x

x

x x

95. 2 6 3 18x x x

First equation: Second equation:

2

2

6 3 18

3 18 0

3 6 0

3 0 3

6 0 6

x x x

x x

x x

x x

x x

2

2

6 3 18

0 9 18

0 3 6

0 3 3

6 6

x x x

x x

x x

x x

x x x

The solutions of the original equation are 3x and 6.x






96. 25 15x x x

First equation: Second equation:

2

2

15 15

16 15 0

1 15 0

1 0 1

15 0 15

x x x

x x

x x

x x

x x

2

2

15 15

14 15 0

1 15 0

1 0 1

15 0 15

x x x

x x

x x

x x

x x

Only 15x and 1x are solutions of the original equation. 1x is extraneous

97. Let 18:y

0.432 10.44

18 0.432 10.44

28.44 0.432

28.44

0.43265.8

y x

x

x

x

x

So, the height of the female is about 65.8 inches or 5 feet 6 inches.

98. Let 21:y

0.449 12.15

21 0.449 12.15

33.15 0.449

33.15

0.44973.8

y x

x

x

x

x

Because the height of the male is about 73.8 inches or 6 feet 2 inches, it is possible the femur belongs to the missing man.

99. False—See Example 14 on page 123.

100. 2 3 1 2 5

2 6 1 2 5

2 5 2 5

x x

x x

x x

False. The equation is an identity, so every real number is a solution.

101. True. There is no value to satisfy this equation.

10 10 0

10 10

10 10

10 10

x x

x x

x x

102. (a) The formula for volume of the glass cube is V = Length × Width × Height. The volume of water in the cube is the length × width × height of the water.

So, the volume is 23 3x x x x x .

(b) Given the equation 2 3 320x x . The

dimensions of the glass cube can be found by solving for x.

So, the capacity of the cube is equal to 3V x .

103. Equivalent equations are derived from the substitution principle and simplification techniques. They have the same solution(s).

2 3 8x and 2 5x are equivalent equations.

Section P.3 The Cartesian Plane and Graphs of Equations

1. Cartesian

2. Distance Formula

3. Midpoint Formula

4. solution or solution point

5. graph

6. intercepts

7. y-axis

8. circle; , ;h k r

9. : 2, 6 , : 6, 2 , : 4, 4 , : 3, 2A B C D

10. 3 52 2

: , 4 , : 0, 2 , : 3, , : 6, 0A B C D

11.


INSTRUCTOR USE ONLY 8.. circle; circle , ;, ;h k rh k r, ;, ;



Section P.3 The Cartesian Plane and Graphs of Equations 15


12.

13. 3, 4

14. 12, 0

15. 0x and 0y in Quadrant IV.

16. 4x and 0y in Quadrant II.

17. ,x y is in the second Quadrant means that ,x y is

in Quadrant III.

18. , , 0x y xy means x and y have the same signs.

This occurs in Quadrant I or III.

19.

Year 3 2003t

20.

21. 2 2

2 1 2 1

2 2

2 2

3 2 6 6

5 12

25 144

13 units

d x x y y

22. 2 2

2 1 2 1

2 2

2 2

0 8 20 5

8 15

64 225

289

17 units

d x x y y

23. 2 2

2 1 2 1

2 2

2 2

5 1 1 4

6 5

36 25

61 units

d x x y y

Month, x Temperature, y

1 –39

2 –39

3 –29

4 –5

5 17

6 27

7 35

8 32

9 22

10 8

11 –23

12 –34

Year, x Number of Stores, y

2003 4906

2004 5289

2005 6141

2006 6779

2007 7262

2008 7720

2009 8416

2010 8970

Tem

pera

ture

(in

°F)

Month (1 ↔ January)

x

−40

10

−30

−20

−10

20

30

2 6 8 10 12

40

0

y

NOT FOR SALESection P.3 The Cartesian ction P.3 The Cartesian Plane and Graphs Plane and Graphs





24. 2 2

2 1 2 1

22

2 2

3.9 9.5 8.2 2.6

13.4 10.8

179.56 116.64

296.2

17.21 units

d x x y y

25. (a) 1, 0 , 13, 5

2 2

2 2

Distance 13 1 5 0

12 5 169 13

13, 5 , 13, 0

Distance 5 0 5 5

1, 0 , 13, 0

Distance 1 13 12 12

(b) 2 2 25 12 25 144 169 13

26. (a) The distance between 1, 1 and 9, 1 is 10.

The distance between 9, 1 and 9, 4 is 3.

The distance between 1, 1 and 9, 4 is

2 2

9 1 4 1 100 9 109.

(b) 2

2 210 3 109 109

27. 2 2

1

2 22

2 23

4 2 0 1 4 1 5

4 1 0 5 25 25 50

2 1 1 5 9 36 45

d

d

d

2 2 2

5 45 50

28. 2 2

1

2 22

2 23

3 1 5 3 16 4 20

5 3 1 5 4 16 20

5 1 1 3 36 4 40

d

d

d

2 2 2

20 20 40

29. 2 2

1

2 22

2 23

1 2

1 3 3 2 4 25 29

3 2 2 4 25 4 29

1 2 3 4 9 49 58

d

d

d

d d

30. 2 2

1

2 22

2 23

1 2

4 2 9 3 4 36 40

2 4 7 9 36 4 40

2 2 3 7 16 16 32

d

d

d

d d

31. (a)

(b) 2 2(5 ( 3)) (6 6) 64 8d

(c) 6 6 5 ( 3)

, (6, 1)2 2

32. (a)

(b) 2 2

9 1 7 1 64 36 10d

(c) 9 1 7 1

, 5, 42 2






33. (a)

(b) 2 2

5 1 4 2

36 4 2 10

d

(c) 1 5 2 4

, 2, 32 2

34. (a)

(b) 2 2

1 5 41

2 2 3

1 829

9 3

d

(c) 5 2 1 2 4 3 1 7

, 1,2 2 6

35. 2 2120 150

36,900

30 41

192.09

d

The plane flies about 192 kilometers.

36. 2 2

2 2

42 18 50 12

24 38

2020

2 505

45

d

The pass is about 45 yards.

37. 2002 2010 19,564 35,123

midpoint ,2 2

2006, 27,343.5

In 2006, the sales for the Coca-Cola Company were about $27,343.5 million.

38. 1 2 1 2midpoint ,2 2

2008 2010 1.89 2.83,

2 2

2009, 2.36

x x y y

In 2009, the earnings per share for Big Lots, Inc. were about $2.36.

39. (a) ?

0, 2 : 2 0 4

2 2

Yes, the point is on the graph.

(b) ?

?

5, 3 : 3 5 4

3 9

3 3


40. (a) ?

?

1, 5 : 5 4 1 2

5 4 1

5 3

No, the point is not on the graph.

(b) ?

?

6, 0 : 0 4 6 2

0 4 4

0 0


41. (a) ?2

?

2, 0 : 2 3 2 2 0

4 6 2 0

0 0


(b) ?2

?

2, 8 : 2 3 2 2 8

4 6 2 8

12 8


42. (a) ?

1, 2 : 2 1 2 3 0

3 0


(b) ?

?

1, 1 : 2 1 1 3 0

2 1 3 0

0 0


NOT FOR SALESection P.3 The Cartesian Plane and Graphs ction P.3 The Cartesian Plane and Graphs

INSTRUCTOR USE ONLY about $27,343.5 million. about $27,343.5 million.





43. (a) ?2 2

?

3, 2 : 3 2 20

9 4 20

13 20


(b) ?2 2

?

4, 2 : 4 2 20

16 4 20

20 20


44. (a) ?3 2

?

?

?

16 1613 3 3

1613 3

8 163 3

8 16243 3 3

16 163 3

2, : 2 2 2

8 2 4

8


(b) ?3 2

?

?

13

13

3, 9 : 3 2 3 9

27 2 9 9

9 18 9

27 9


45. 2 5y x

46. 34

1y x

47. 2 3y x x

48. 25y x

x 1 0 1 2 52

y 7 5 3 1 0

(x, y) 1, 7 0, 5 1, 3 2, 1 52, 0

x 2 0 1 43

2

y 52

–1 14

0 12

(x, y) 52

2, 0, 1 14

1, 43, 0 1

22,

x 1 0 1 2 3

y 4 0 –2 –2 0

(x, y) 1, 4 0, 0 1, 2 2, 2 3, 0

x 2 –1 0 1 2

y 1 4 5 4 1

x, y 2, 1 1, 4 0, 5 1, 4 2, 1

requisitesrequisites





49. 216 4y x

x-intercepts: 2

2

2

0 16 4

4 16

4

2

x

x

x

x

2, 0 , 2, 0

y-intercept: 2

16 4 0 16

0, 16

y

50. 2

3y x

x-intercept: 2

0 3

0 3

3

x

x

x

3, 0

y-intercept: 2

2

0 3

3

9

0, 9

y

y

y

51. 5 6y x

x-intercept:

65

0 5 6

6 5

x

x

x

65, 0

y-intercept: 5 0 6 6y

0, 6

52. 8 3y x

x-intercept:

83

0 8 3

3 8

x

x

x

83, 0


0, 8

53. 4y x

x-intercept: 0 4

0 4

4

x

x

x

4, 0

y-intercept: 0 4 2y

0, 2

54. 2 1y x

x-intercept:

12

0 2 1

2 1 0

x

x

x

12, 0

y-intercept: 2 0 1

1

y

There is no real solution. There is no y-intercept.

55. 3 7y x

x-intercept:

73

0 3 7

0 3 7

0

x

x

73, 0


0, 7

56. 10y x

x-intercept: 0 10

10 0

10

x

x

x

10, 0

y-intercept: 0 10

10 10

y

0, 10






57. 3 22 4y x x

x-intercept: 3 2

2

0 2 4

0 2 2

0 or 2

x x

x x

x x

0, 0 , 2, 0

y-intercept: 3 2

2 0 4 0

0

y

y

0, 0

58. 4 25y x

x-intercepts: 4

4

4 2

0 25

25

5 5

x

x

x

5, 0

y-intercept: 4

0 25 25y

0, 25

59. 2 6y x

x-intercept: 0 6

6

x

x

6, 0

y-intercepts: 2 6 0

6

y

y

0, 6 , 0, 6

60. 2 1y x

x-intercept: 0 1

1

x

x

1, 0

y-intercepts: 2 0 1

1

y

y

0, 1 , 0, 1

61.

62.

63.

64.

65. 2 0x y

2 2

2 2

2 2

0 0 -axis symmetry

0 0 No -axis symmetry

0 0 No origin symmetry

x y x y y

x y x y x

x y x y






66. 2 0x y

2 2

2 2

2 2


0 0 -axis symmetry


x y x y y

x y x y x

x y x y

67. 3

3 3

3 3

3 3 3

No -axis symmetry

No -axis symmetry

Origin symmetry

y x

y x y x y

y x y x x

y x y x y x

68. 4 2

4 2 4 2

4 2 4 2

4 2 4 2

3

3 3 -axis symmetry



y x x

y x x y x x y

y x x y x x x

y x x y x x

69. 2

2 2

2 2

2 2 2

1

No -axis symmetry11

No -axis symmetry1 1

Origin symmetry1 11

xy

xx x

y y yxx

x xy y x

x xx x x

y y yx xx

70. 2

2 2

2 2

2 2

1

11 1

-axis symmetry11

1 1No -axis symmetry

1 11 1

No origin symmetry11

yx

y y yxx

y y xx x

y yxx

71. 2

2 2

2 2

2 2

10 0

10 0 10 0 No -axis symmetry

10 0 10 0 -axis symmetry

10 0 10 0 No origin symmetry

xy

x y xy y

x y xy x

x y xy

72. 4



4 4 Origin symmetry

xy

x y xy y

x y xy x

x y xy






73. 3 1y x

x-intercept: 13, 0

y-intercept: 0, 1

No symmetry

74. 2 3y x

x-intercept: 32, 0

y-intercept: 0, 3

No symmetry

75. 2 2y x x

x-intercepts: 0, 0 , 2, 0

y-intercept: 0, 0

No symmetry

76. 2 1x y

x-intercept: 1, 0

y-intercepts: 0, 1 , 0, 1

x-axis symmetry

77. 3 3y x

x-intercept: 3 3, 0

y-intercept: 0, 3

No symmetry

78. 3 1y x

x-intercept: 1, 0

y-intercept: 0, 1

No symmetry

x –1 0 1 2 3

y 3 0 –1 0 3

x –1 0 3

y 0 ±1 ±2

x –2 –1 0 1 2

y –5 2 3 4 11






79. 3y x

x-intercept: 3, 0

y-intercept: none

No symmetry

80. 1y x

x-intercept: 1, 0

y-intercept: 0, 1

No symmetry

81. 6y x

x-intercept: 6, 0

y-intercept: 0, 6

No symmetry

82. 1y x

x-intercepts: 1, 0 , 1, 0

y-intercept: 0, 1

y-axis symmetry

83. Center: 0, 0 ; Radius: 4

2 2 2

2 2

0 0 4

16

x y

x y

84. Center: 7, 4 ; Radius: 7

2 2 2

2 2

7 4 7

7 4 49

x y

x y

85. Center: 1, 2 ; Solution point: 0, 0

2 2 2

2 2 2 2

2 2

1 2

0 1 0 2 5

1 2 5

x y r

r r

x y

86. Center: 3, 2 ; Solution point: 1,1

2 2

22

3 1 2 1

4 3 25 5

r

22 2

2 2

3 2 5

3 2 25

x y

x y

87. Endpoints of a diameter: 0, 0 , 6, 8

Center: 0 6 0 8

, 3, 42 2

2 2 2

2 2 2 2

2 2

3 4

0 3 0 4 25

3 4 25

x y r

r r

x y

x 3 4 7 12

y 0 1 2 3

x –2 0 2 4 6 8 10

y 8 6 4 2 0 2 4

Section P.3 The Cartesian Plane and Graphs ction P.3 The Cartesian Plane and Graphs





88. Endpoints of a diameter: 4, 1 , 4,1

2 2

2 2

14 4 1 1

21

8 221

64 42

1 168 2 17 17

2 2

r

Midpoint of diameter (center of circle):

22 2

2 2

4 4 1 1, 0, 0

2 2

0 0 17

17

x y

x y

89. 2 2 25x y

Center: 0, 0 , Radius: 5

90. 22 1 1x y


91. 2 2 91 1

2 2 4x y

Center: 1 12 2, , Radius: 3

2

92. 2 2 16

92 3x y


93. 500,000 40,000 , 0 8y t t

94. 8000 900 , 0 6y t t






95. (a)

(b)

When 85.5,x the resistance is about 1.4 ohms.

(c) When 85.5,x

2

10,3701.4 ohms.

(85.5)y

(d) As the diameter of the copper wire increases, the resistance decreases.

96. (a)

Because the line is close to the points, the model fits the data well.

(b) Graphically: The point 90, 75.4 represents a life expectancy of 75.4 years in 1990.

Algebraically: 220.002 0.5 46.6 0.002 90 0.5 90 46.6 75.4 y t t

So, the life expectancy in 1990 was about 75.4 years.

(c) Graphically: The point 94.6, 76.0 represents a life expectancy of 76 years during the year 1994.

Algebraically: 2

2

2

0.002 0.5 46.6

76.0 0.002 0.5 46.6

0 0.002 0.5 29.4

y t t

t t

t t

Use the quadratic formula to solve.

2

2

4

2

0.5 0.5 4 0.002 29.4

2 0.002

0.5 0.0148

0.004

125 30.4

b b act

a

So, 94.6t or 155.4.t Since 155.4 is not in the domain, the solution is 94.6t , which is the year 1994.

(d) When 115:t

220.002 0.5 46.6 0.002 115 0.5 115 46.6 77.65y t t

The life expectancy using the model is 77.65 years, which is slightly less than the given projection of 78.9 years.

(e) Answers will vary. Sample answer: No. Because the model is quadratic, the life expectancies begin to decrease after a certain point.

x 5 10 20 30 40 50 60 70 80 90 100

y 414.8 103.7 25.9 11.5 6.5 4.1 2.9 2.1 1.6 1.3 1.0


INSTRUCTOR USE ONLY after a certain point.after a certain point.



Education

Solution manual-for-trigonometry-9th-edition-by-larson