Chapter 5orensclass.weebly.com/uploads/1/2/3/5/12353620/chapter_5-8_odd_answers.pdf · 75. 77. 79. 81. 83. 85. 87. 89. 91. 93. 95. 97. 99. 101. The identity is not true when 103

2. 2400 radians per minute

3. sin 4. sin

cos cos

tan csc

csc sec

sec cot

cot

5.

6. Quadrant III 7. 8. 1.33, 1.81

9. sin

tan

csc

sec

cot

10. 11.

12. 13.

14. 15.

16. 17.

Period: 2 Not periodic

18. 19.

20. 21.

22. 23.

Chapter 5Section 5.1 (page 357)

Vocabulary Check (page 357)1. 2. 3. 4.

5. 6. 7. 8.

9. 10.

1. 3.

cot ! " #1cot x " !3

csc ! " #!2sec x "2!3

3

tan ! " #1csc x " 2

cos ! "!22

tan x "!33

cos u#tan u

sec usin ucsc2 utan2 u

csc ucot utan usec u

214.51$

−10 10

%2

%2

−

2−20

2%

3−3

%−

%

!52

c " #%4

b "12

,a " #2,

0

−2

6

32

−4

4

6−6

4

x

y

−3

−4

π2

π2

3π2

−π π2π−

2

π2−−2

−4

x

y

4

1

5

− π2

π2

π−π

6

3

x

y

ππ−

2

1

4

3

x

y

1

2

x

y

π2

π4

–4

–3

–2

–1

1

3

4

xπ2

y

! " #34

! " #53

! " 54

! " #43

! " 45

135$, 225$

x

y

255$=θ

= 75°θ ′

!& " 75$

! " #14

! "27

! " #!17

! "!53

2! "

!174

! "!53

7! " #4

! "2!53

53! " #

!1717

! "7!53

53! "

4!1717

Answers to Odd-Numbered Exercises and Tests A205

CH

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333350_05_ans_odd.qxp 1/15/07 3:29 PM Page A205

x 0.2 0.4 0.6 0.8

y1 1.2230 1.5085 1.8958 2.4650

y2 1.2230 1.5085 1.8958 2.4650

x 1.0 1.2 1.4

y1 3.4082 5.3319 11.6814

y2 3.4082 5.3319 11.6814

x 0.2 0.4 0.6 0.8

y1 0.1987 0.3894 0.5646 0.7174

y2 0.1987 0.3894 0.5646 0.7174

x 1.0 1.2 1.4

y1 0.8415 0.9320 0.9854

y2 0.8415 0.9320 0.9854

5. 7.

9. 11.

13.

is undefined.

15. d 16. a 17. b 18. f 19. e

20. c 21. b 22. c 23. f 24. a

25. e 26. d 27. 29.

31. 33. 1 35. 37.

39–49. Answers will vary. 51. 53.

55. 57. 59.

61. 63. 65.

67. 69. 71.

73.

75.

77. 79. 81. 83.

85. 87. 89. 91.

93. 95.

97. 99.

101. The identity is not true when



107. (a) and (b) Answers will vary.

109. (a) and (b) Answers will vary.

111. Answers will vary. 113. True, for all

115. 1, 1 117.

119.

The sign depends on the choice of

121. Answers will vary.

!.

cot ! " ±!1 # sin2 !

sin !

sec ! " ±1

!1 # sin2 !

csc ! "1

sin !

tan ! " ±sin !

!1 # sin2 !

cos ! " ±!1 # sin2 !

', 0

! ( n%

! "7%4

.

! "5%3

.

! "7%6

.

ln"#cos x$#1 ) sin x$"ln"cot !"0 ≤ ! <

%2

, 3%2

< ! < 2%0 ≤ ! ≤ %

!2 cos !3 tan !3 sec !3 cos !

3 tan !5 cos !tan xcsc x

00

1.5

12

y1 = y2

00

1.5

1.0

y1 = y2

3#sec x ) tan x$1 ) cos y#cot x

2 csc2 xcot2 x1 ) 2 sin x cos x

#csc x # 1$ cot2 xsin2 x # cos2 xsec4 x

cos x ) 2cos2 x

1 ) sin ycot xsec x

cos2 *cos x

cot !

sec ! " #1

tan ! " 0

cos ! " #1

sin ! " 0

cot ! "12

cot x " #!52

sec ! " #!5sec x " #3!5

5

csc ! " #!52

csc x "32

cos ! " #!55

cos x " #!53

sin ! " #2!5

5sin x "

23

cot * " #158cot x " 24

7

csc * " 178csc x " #25

7

tan * " # 815cos x " #24

25

cos * " #1517sin x " # 7

25

A206 Answers to Odd-Numbered Exercises and Tests


x 0.2 0.4 0.6 0.8

y1 5.1359 2.7880 2.1458 2.0009

y2 5.1359 2.7880 2.1458 2.0009

x 1.0 1.2 1.4

y1 2.1995 2.9609 5.9704

y2 2.1995 2.9609 5.9704

x 0.2 0.4 0.6 0.8

y1 5.0335 2.5679 1.7710 1.3940

y2 5.0335 2.5679 1.7710 1.3940

x 1.0 1.2 1.4

y1 1.1884 1.0729 1.0148

y2 1.1884 1.0729 1.0148

x 0.2 0.4 0.6 0.8

y1 4.8348 2.1785 1.2064 0.6767

y2 4.8348 2.1785 1.2064 0.6767

x 1.0 1.2 1.4

y1 0.3469 0.1409 0.0293

y2 0.3469 0.1409 0.0293

x 0.2 0.4 0.6 0.8

y1 4.8348 2.1785 1.2064 0.6767

y2 4.8348 2.1785 1.2064 0.6767

x 1.0 1.2 1.4

y1 0.3469 0.1409 0.0293

y2 0.3469 0.1409 0.0293

123. 125.

Section 5.2 (page 365)

Vocabulary Check (page 365)1. conditional 2. identity 3. 4.

5. 6. 7. 8.

9. 10.

1– 9. Answers will vary.

11.

13.

15.

17.

19.

21–49. Answers will vary.

51. 53.

55. Answers will vary. 57. Answers will vary.

59. 1 61. 2 63–69. Answers will vary.

71. 73. True

75. False.

77. (a) Answers will vary. (b) No. Division by zero.

79. (a) Answers will vary. (b) Yes. Answers will vary.

sin2%%4& ) cos2%%

4& ( 1 ) tan2%%4&

W ( 0+ " tan !,

y " sin xy " 1

−2

2

60

−2

2

3−3

cot##x$ " #cot#x$

00

1.5

6

y1 = y2

00

1.5

5

y1 = y2

00

1.5

5

y1 = y2

00

1.5

5

y1 = y2

sec u#sin u

cot ucos2 ucos utan u

sin ucot u

4

π−π−1

−2

x

y

1

−1

1

13 1 3

2

22 2 2−−

x

y


CH

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81. 83. 85.

87. Answers will vary. 89.

91.

93. 95.

97. Quadrant II 99. Quadrant IV


Vocabulary Check (page 376)1. general 2. quadratic 3. extraneous


11. 13. 15.

17. 19. 21. 23.

25. 27.

29. 31.

33. 35.

37. 39.

41. 43. No solution 45.

47. 49. 3.6652, 4.7124, 5.7596

51. 0.8614, 5.4218 53. 1.5708 55. 0.5236, 2.6180

57. (a)

(b)

(c)

59. (a)

(b)

(c)

61. 63. 65.

67. 69.

71. 73.

75. 77. 79. 0.8603, 3.4256

81. 0, 2.6779, 3.1416, 5.8195

83. 0.3398, 0.8481, 2.2935, 2.8018

85. 87. 1.110

89. (a)

Maxima:

Minima:

(b)

91. (a)

Maxima:

Minima:

(b)

93. (a)

Maximum:

Minimum:

(b)%4

, 5%4

#3.9270, #1.4142$#0.7854, 1.4142$

0

−2

2

%2

0, %3

, %, 5%3

, 2%

#0, 1$, #3.1416, #1$, #6.2832, 1$#5.2360, 1.25$#1.0472, 1.25$,

0 π2

−2

2

%4

, 3%4

, 5%4

, 7%4

#5.4978, #1$#2.3562, #1$,#3.9270, 1$#0.7854, 1$,

0 π2

−2

2

#1.154, 0.534

1.1071, 4.2487x " ±2

x " #1, 3%2

) 4n%, 7%2

) 4n%

n%3

, %4

) n%%8

)n%4

2%3

) n%, 5%6

) n%%8

)n%2

2% ) 4n%

#2.2680, 0.5878$#0.3194, 0.0986$,sin2 x " ex # 4x

0 π2

−2

4

#1.7757, #0.3984$#0, 0$,sin 2x " x2 # 2x

0 π2

−2

4

2.0344, 5.1760, %4

, 5%4

%3

, 5%3

%3

, %, 5%3

0, %6

, 5%6

, %, 7%6

, 11%

6%4

, 3%4

, 5%4

, 7%4

2%3

, 5%3

%3

) n%, 2%3

) n%

%3

) n%, 2%3

) n%%3

) n%, 2%3

) n%

%6

) 2n%, 11%

6) 2n%

2%3

) 2n%, 4%3

) 2n%

3%4

, 7%4

%6

, 7%6

7%6

, 11%

65%6

, 11%

6

3%4

, 7%4

5%6

, 7%6

45$, 225$

120$, 240$30$, 150$

−1−2−3−4 1 2 3 4−1

1

2

3

4

5

6

7

y

x

−2−4

2

6

8

2 4x

y

f #x$ " x3 # 16x2 ) 85x # 148

f #x$ " x3 # x2 ) 64x # 64

!tan2 x " "tan x"; 3%4

a sec !a cos !



x 0.2 0.4 0.6 0.8

y1 0.6621 0.7978 0.9017 0.9696

y2 0.6621 0.7978 0.9017 0.9696

x 1.0 1.2 1.4

y1 0.9989 0.9883 0.9384

y2 0.9989 0.9883 0.9384

95. 1

97. (a) All real numbers x except

(b) axis symmetry; horizontal asymptote:

(c) Oscillates

(d) Infinite number of solutions

(e) Yes. 0.6366

99. April, May, June, July

101. 0.04 second, 0.43 second, 0.83 second

103.

105. (a)

(b) By graphing the curves, it can be seen that model (1),best fits the data.

(c) The constant terms gives the average unemploymentrate of 5.45%.

(d) 13.37 years

(e) 2010

107. False

109. False. The range of the sine function does not include 3.4.

111. 2.164 rad 113. rad

115. 24.25 117. 2290.38 feet


Vocabulary Check (page 384)1.

2. 3.

4.

5. 6.

1. (a) (b) 3. (a) (b)

5. (a) (b)

7. 9.

11. 13.

15. 17.

19.

21.

23. 25. 27. 29.

31.

00

1.5

2

y1 = y2

cos 16%63

sin 2.3tan 239$cos 80$

tan%#7%12& " 2 ) !3

cos%#7%12& "

!2 # !64

sin%#7%12& " #

!2 ) !64

tan 13%12

" 2 # !3

cos 13%12

" #!2 ) !6

4

sin 13%12

"!2 # !6

4

tan##225$$ " #1tan 75$ " 2 ) !3

cos##225$$ " #!22

cos 75$ "!6 # !2

4

sin##225$$ "!22

sin 75$ "!2 ) !6

4

tan%#%12& " !3 # 2tan

11%12

" #2 ) !3

cos%#%12& "

!2 ) !64

cos 11%12

" #!2 ) !6

4

sin%#%12& "

!2 # !64

sin 11%12

"!6 # !2

4

tan 195$ " 2 # !3tan 105$ " #2 # !3

cos 195$ " #!2 ) !6

4cos 105$ "

!2 # !64

sin 195$ "!2 # !6

4sin 105$ "

!2 ) !64

#!2 ) !3

2#!2 ) !6

4

1 ) !22

!2 # !64

#32

#12

tan u # tan v1 ) tan u tan v

cos u cos v ) sin u sin v

sin u cos v ) cos u sin v

tan u ) tan v1 # tan u tan v

cos u cos v # sin u sin v

sin u cos v # cos u sin v

#0.007

r " 124 sin#0.47t ) 0.40$ ) 5.45,

2 4 6 8 10 12 14

1

2

3

4

5

6

7

8

t

r

Une

mpl

oym

ent r

ate

Year (0 1990)↔

36.87$, 53.13$

y " 1y-

x " 0


CH

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x 0.2 0.4 0.6 0.8

y1 0.9605 0.8484 0.6812 0.4854

y2 0.9605 0.8484 0.6812 0.4854

x 1.0 1.2 1.4

y1 0.2919 0.1313 0.0289

y2 0.2919 0.1313 0.0289

33.

35. 37. 39. 41. 43. 1

45. 47. 1 49. 51.

53. 55. 0 57. 0 59. 61.


75. 0.7854, 5.4978 77. 0, 3.1416


81. False.

83. Answers will vary. 85. Answers will vary.

87. (a) (b)

89. (a) (b)

91. 93. Answers will vary.


99. 101. 103.


Vocabulary Check (page 394)

1. 2. 3.

4. 5.

6. 7.

8. 9.

10.

1. (a) (b) (c) (d)

(e) (f) (g) (h)

3.

5. 0.2618, 1.3090, 3.4034, 4.4506;

7. 0, 2.0944, 4.1888;

9. 0, 1.5708, 3.1416, 4.7124;

11. 1.5708, 3.6652, 5.7596;

13. 15.

17.

19. 21.

23. 25.

27.

29. 31.

33. 35.

37. (a) (b) (c) (d)

(e) (f) 4 (g) (h)

39. 41.

43. 45.

tan 3%8

" !2 ) 1tan %8

" !2 # 1

cos 3%8

"!2 # !2

2cos

%8

"!2 ) !2

2

sin 3%8

"!2 ) !2

2sin

%8

"!2 # !2

2

tan 112$ 30& " #1 # !2tan 15$ " 2 # !3

cos 112$ 30& " #!2 #!2

2cos 15$ "

!2 ) !32

sin 112$ 30& "!2 )!2

2sin 15$ "

!2 # !32

2!1717

817

!17

!174

14

!1717

4!1717

18#3 # 4 cos x ) cos 2x$1

8#1 # cos 8x$

12#1 ) cos x$1

2#1 # cos 4x$

132#2 ) cos 2x # 2 cos 4x # cos 6x$

18#1 # cos 4x$1

8#3 ) 4 cos 2x ) cos 4x$6 cos 2x4 sin 2x

tan 2u "4!21

17

cos 2u " #1725

sin 2u " #4!21

25

tan 2u " 43tan 2u " 24

7

cos 2u " 35cos 2u " 7

25

sin 2u " 45sin 2u " 24

25

%2

, 7%6

, 11%

6

0, %2

, %, 3%2

0, 2%3

, 4%3

%12

, 5%12

, 13%12

, 17%12

0, 1.0472, 3.1416, 5.2360; 0, %

3, %,

5%

3

724

2524

257

247

2425

725

45

35

2 sin%u ) v2 & cos%u # v

2 &

12

'sin#u ) v$ ) sin#u # v$(cos u2

sin2 u12'cos#u # v$ ) cos#u ) v$(

2 tan u1 # tan2 u

tan u2

cos 2u1 ) cos 2u

22 sin u cos u

%2

%6

#0, 4$, #2, 0$, #7, 0$

#0, 19$, #38, 0$u ) v " w.

2 cos !

13 cos#3! # 1.1760$13 sin#3! ) 0.3948$

!2 cos%! #%4&!2 sin%! )

%4&

cos#u ± v$ " cos u cos v , sin u sin v

0, %3

, %, 5%3

%2

2425

3365

!33

#12

#12x2 # !1 # x2

!4x2 ) 1

1385

3685#63

16#6365

00

1.5

2

y1 = y2



47. 49.

51.

53. 55. 57.

59. 61.

63. 65.

67. 69.

71. 73.

75. 77.

79. 81. 83.

85. 87. 89.

91. 93–109. Answers will vary.

111. 113.

115. 117. 119.

121. (a)

Maximum: (3.1416, 3)

(b)

123. (a)

Minima:

Maxima:

(b) 0.6993, 2.6078, 3.6754, 5.5839

125. (a)

(b)

127. (a)

(b) feet

(c) feet per second

(d) Answers will vary.

129.

131. False. for

133. (a) (b)

(c) Answers will vary.


137. (a) (b)

(c)

139. (a) (b)

(c) #23, 32$

2!133

x

y

−2 −1

−1

1 2 3

1

2

3

−2

12

0,( (

( (43

52

,

#2, 3$2!10

−2

−11 2 3 4 5 6−1−2

−3

−4

1

2

3

4

x

y

(5, 2)

(−1, 4)

y " sin 2x

−2

−2

2

2

%%

% ≤x2≤ 2%.sin

x2

" #!1 # cos x2

x " 2r#1 # cos !$! " 45$;

98.73

198.90

r " 116 v0

2 sin ! cos !

2% # arccos 1 ± !33

8x " arccos

1 ± !338

,

2%5%3

,%,%3

,x " 0,

#0.6993, 2.8642$, #3.6754, 0.3486$#2.6078, #0.3486$, #5.5839, #2.8642$

0

−4

4

%2

%

00

4

%2

1 # x2

1 ) x21 # 2x22x!1 # x2

−2

−2

2

2

%%−2

−2

2

2

%%

3 ) 4 cos 2x ) cos 4x8

1 # cos 2x2

413

25169

%6

, 5%6

0, %4

, %2

, 3%4

, %, 5%4

, 3%2

, 7%4

!62

!22

#2 sin ! sin %2

2 cos - sin .2 cos 4x cos 2x

2 cos 3! sin 2!12#sin 2! ) sin 2%$

12#cos 2y # cos 2x$5

2#cos 8. ) cos 2.$5#cos 60$ ) cos 90$$1

2#sin 8! ) sin 2!$

3%sin 2%3

) sin 0&%3

, %, 5%3

%3

, 5%3

#"tan 4x""sin 3x"

tan u2

" #3

cos u2

" #!1010

sin u2

"3!10

10

tan u2

"5 # !89

8tan

u2

" 5

cos u2

" #!89 ) 5!89178

cos u2

"!2626

sin u2

"!89 # 5!89178

sin u2

"5!26

26


CH

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141. (a) Complement: supplement:

(b) Complement: none; supplement:

143. (a) Complement: supplement:

(b) Complement: supplement:

145. 0.4667 rad

147. 149.

Review Exercises (page 400)1. 3. 5. 7. 9.

11.

15. 17. 19. 1 21.


37. 39.

41. 43.

45. 47. 49.

51. 53.

55. 57.

59. 61. 63.

65. arctan 2,

67.

69.

71. 73. 75.

77. 79. 81. 0 83. 0

85–89. Answers will vary. 91.

93. 95.

97 and 99. Answers will vary.

101. 103.

105.

107.

109. 111.

113.

115. 117. cubic metersV " sin !2

cos !2

#"cos 4x"

tan u2

"3!5

5

cos u2

"!7014

sin u2

"3!14

14

tan u2

"13

tan 7%8

" 1 # !2

cos u2

"3!10

10cos

7%8

" #!2 ) !2

2

sin u2

"!1010

sin 7%8

"!2 # !2

2

tan 105$ " #2 # !3

cos 105$ " #!2 # !3

2

sin 105$ "!2 ) !3

2

18#3 ) 4 cos 4x ) cos 8x$

132#10 # 15 cos 2x ) 6 cos 4x # cos 6x$15$, 75$

tan 2u " #3677

tan 2u " #20!6

cos 2u "7785

cos 2u " #149

sin 2u " #3685

sin 2u "20!6

49

%4

, 7%4

# 44125

34

#117125tan 35$sin 180$

tan 31%12

" #2 # !3

cos 31%12

"!2 # !6

4

sin 31%12

"!2 ) !6

4

tan 285$ " #2 # !3

cos 285$ "!6 # !2

4

sin 285$ " #!2 ) !6

4

arctan##5$ ) 2%

arctan 2 ) %,arctan##5$ ) %,

0, %%12

)n%6

%12

) n%, 5%12

) n%

%2

, 3%2

0, %8

, 3%8

, 5%8

, 7%8

, 9%8

, 11%

8,

13%8

, 15%

8

%8

, 3%8

, 9%8

, 11%

80,

%2

, %

0, 2%3

, 4%3

n%%6

) n%, 5%6

) n%

%3

) n%, 2%3

) n%%6

) n%

%3

) 2n%, 2%3

) 2n%%6

) 2n%, 5%6

) 2n%

csc x1 ) cot -cos2 x

cot x "34

sec x "53

csc x "54

tan x "43

sec xsec x"sin x"cos xsec x

4

3

2

1

12

− 12

x

y

−2

−1

2

−π πx

y

11%20

%20

;

17%18

4%9

;

18$

125$35$;


13.

cot x " #1

sec x " !2

csc x " #!2

tan x " #1

cos x "!22



119. 121.

123. 125.

127. 129. feet

131. False. 133. True


Chapter Test (page 403)

1. 2. 1 3. 1

4. 5.

6.


14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.



1. oblique 2.

3. (a) Two; any; AAS; ASA

(b) Two; an opposite; SSA

4.

1. inches

3. centimeters, centimeters

5. kilometers, kilometers

7.

9.

11.

13.

15. No solution

17. Two solutions

19. 28.19 square units 21. 1782.32 square units

23. 2888.57 square units

25. (a)

(b) (c) 9.52 meters

27.

29. (a) (b) 4385.71 feet

(c) 3061.80 feet

31. 15.53 kilometers from Colt Station; 42.43 kilometers fromPine Knob

40°

3000

ft

r

r

s

240.03!

16sin 70!

"h

sin 34!

34°

14°

h 70°

20°

16

B ! 107.79!, C ! 14.21!, c ! 3.30

B ! 72.21!, C ! 49.79!, c ! 10.27

B ! 48.74!, C ! 21.26!, c ! 48.23

B ! 18!13#, C ! 51!32#, c ! 40.05

B " 60.9!, b ! 19.32, c ! 6.36

B ! 21.55!, C ! 122.45!, c ! 11.49

c ! 6.26a ! 6.41C " 74!15#,

b ! 6.32A " 40!, a ! 15.69

C " 95!, b ! 24.59 inches, c ! 28.29

12ac sin B1

2ab sin C;12bc sin A;

bsin B

tan 2u " $43

cos 2u " $35

76.52!sin 2u " 45

$2.938, $2.663, 1.170%6

, 5%6

, 3%2

%6

, 5%6

, 7%6

, 11%

6%6

, %2

, 5%6

, 3%2

0, 3%4

, %, 7%4

$2 cos 7&2

sin &2

2"sin 6& ' sin 2&#tan 2&

116$

10 $ 15 cos 2x ' 6 cos 4x $ cos 6x1 ' cos 2x %

$2 $ &3

y1 " y2

π2π2

−4

4

−

0, %2

< & ≤ %, 3%2

< & < 2%csc & sec &

cot & "23

sec & "$&13

2

csc & "$&13

3

cos & "$2&13

13

sin & "$3&13

13

y3 " y2 ' 1

cos &2

> 0

&102

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%4

2 cos'9&2 ( cos'&

2(

12

"cos ( $ cos 9(#3'sin %2

' sin 0(

CH

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R 6

333350_06_ans_odd.qxp 1/19/07 8:59 AM Page A213

33.

35. (a)

(b)

(c)

(d)

37. False. The triangle can’t be solved if only three angles are known.

39. Yes, the Law of Sines can be used to solve a right triangleprovided that at least one side and one angle are given or twosides are given. Answers will vary.

41.

43.

45. 47.


Vocabulary Check (page 421)1. 2. Heron’s Area

3.

1.

3.

5. millimeters

7. feet

9.

11.

13.

15.

17.

19.

21. 4 8 11.64 4.96

23. 10 14 20 13.86

25. 15 16.96 25 20

27. 104.57 square inches 29.

31. 0.27 square foot 33. 15.52 35. 35.19

37.

1357.85 miles,

39.

41. 43.27 miles 43.

45. square feet

47. (a)

(b)

(c)

(d) 6 inches

49. False. A triangle cannot be formed with sides of lengths 10 feet, 16 feet, and 5 feet.

51. False. 53 and 55. Proof.

57. Answers will vary. Yes; Answers will vary.

59. 61.


Vocabulary Check (page 433)1. directed line segment 2. initial, terminal

3. magnitude 4. vector

5. standard position 6. unit vector

7. multiplication, addition 8. resultant

9. linear combination, horizontal, vertical

%3

$%2

s "a ' b ' c

2

00 2%

10

x " 12"3 cos & ' &9 cos2& ' 187 #

49 " 2.25 ' x 2 $ 3x cos &

18,617.66

PQ ! 9.43, QS ! 5, RS ! 12.81

N 43.03! E, S 66.95! E

AB 3600 meters

1500 meters 2800 meters

C S

W E

N

236.01!

S

W E

Franklin

Centerville

Rosemont

75° 32°

810 mi

648 mi

N

19.81

102.78!77.22!

111.80!68.20!

150!30!

)&dcba

A ! 37!6# 7*, C ! 67!33#53*, b ! 9.94

A ! 154!14#, C ! 17!31#, b ! 8.58

A ! 103.52!, B ! 38.24!, C ! 38.24!

A ! 36.87!, B ! 53.13!, C " 90!

B ! 29.44!, C ! 100.56!, a ! 23.38

A ! 26.38!, B ! 36.34!, C ! 117.28!

A ! 53!45#, C ! 75!45#, b ! 9.95

A ! 31.40!, C ! 128.60!, b ! 6.56

A ! 49.51!, B ! 55.40!, C ! 75.09!

A ! 40.80!, B ! 60.61!, C ! 78.59!

12bh, &s"s $ a#"s $ b#"s $ c#c2 " a2 ' b2 $ 2ab cos C

32'sin

11%6

' sin 3%2 (3"sin 11& ' sin 5&#

cot & " $ 512sec & " 13

5 ;csc & " $1312;tan & " $12

5 ;

C " 83!, b ! 17.83, c ! 22.46

d " sin"84.64 $ &#$58.36sin & %

+ " arcsin'd sin &58.36 (

( ! 5.36!

16.08!


& 10! 20! 30! 40! 50! 60!

d 324.08 154.19 95.19 63.80 43.30 28.10



3. 5.

7. 9.

11.

13. 15.

17. 19.

21. 23.

25. (a) (b) (c) (d)

27. (a) (b) (c)

(d)

29. (a) (b) (c)

(d)

31. 33. 35.

37. 39. 41.

43. 45. 47.

49. 51. 53.

55. 57.

59. 61.

63. 65.

67. 69.

71. 73.

75. 77. 90!)10&2 $ 25&3, 25 ' 10&2 *

≈ 71.57°θ

10 1035 5

,( )

−4 −3 −2 −1 4

−4

−3

−2

1

2

3

4

31 2x

y

+52

, 10 ' 5&3

2 ,v " +&105

, 3&10

5 ,

= 150°θ

63 3 22 2

,( )

− 4 −3 −2 −1 4

− 4

−3

−2

2

31 2

−

x

y

−4 −3 −2 −1 4

−4

−3

−2

3

4

1

2

31 2

= 0° (3, 0)θx

y

v " +$3&62

, 3&2

2 ,v " )3, 0*

-v- " &29, & " 111.80!-v- " 6&2, & " 315!

−1 4

−1

−2

1

2

y

x

12

w

32

u

12

(3u + w)

-v- " 5, & " 30!v " )72, $1

2*

x

u

2w

u w+ 2

3 4 5

−1

1

2

3

4

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1

1

2 3

−1

−2

u32 u

x

y

v " )4, 3*v " )3, $32*

3i ' 8j7i ' 4j$8i

215

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j40&61

61i '

48&6161

jj

45

i $35

j+$2425

, $7

25,+$&22

, &22 ,

)1, 0*w $ vu ' v

6i ' j

$4i ' 11j$i ' 4j3i $ 2j

)$22, $28*

)$18, $28*)$8, $12*)$4, $4*

)23, 9*)$13, 1*)$3, 1*)11, 3*

x

y

uv

v − u12

12−

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y

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u + 2v

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y

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, $35,,)0, 5* , - v - " 5

)$3, 2* , - v - " &13)4, 3* , - v - " 5


CH

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R 6


& 0! 30! 60! 90!

M 370! 357.85! 322.34! 266.27!

( 0! 12.10! 23.77! 34.29!

& 120! 150! 180!

M 194.68! 117.23! 70!

( 41.86! 39.78! 0!

& 10! 20! 30!

T 3046.28 3192.53 3464.10

& 40! 50! 60!

T 3916.22 4667.17 6000

79.

81.

83. Horizontal component: feet per second

Vertical component: feet per second

85.

87. (a) Domain:

(b)

(c)

(d) The component in the direction of the motion of thebarge decreases.

89. 130.35 kilometers per hour

91. (a) 357.85 newtons

(b)

(c)

(d)

(e) For increasing the two vectors tend to work againsteach other, resulting in a decrease in the magnitude ofthe resultant.

93. True 95. True. 97. True

99. True 101. True 103. False

105. (a) (b)

(c) No. The magnitude is equal to the sum when the anglebetween the vectors is

107. Proof. 109.

111. 113.

115. 117. 119.

121. 123.



1. dot product 2. 3. orthogonal

4. 5.

1. 0 3. 14 5. 8, scalar 7. 4, scalar

9. vector 11. 13 13.

15. 4 17. 19. 21. 23.

25. 27.

29. 31. 33. Parallel

35. Neither 37. Orthogonal 39. 3

41. 10 43. 0

$162&226.57!, 63.43!, 90!

& ! 13.57!& ! 4.40!

−2

−4

−6

−8

−10

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2

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v

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−2

−3

−4

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−6

1

1

2 3 4 5 6x

y

uv

5%12

90!70.56!90!

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-projPQ\ F - -PQ

\

-, F , PQ\'u , v

-v-2 (v

u , v-u- -v-

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' 2n%%2

' n%, % ' 2n%

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48xy2, x . 012x3y7, x . 0, y . 0

)1, 3* or )$1, $3*0!.

180!0!

a " b " 0

&,

00 180

45

00 180

500

( " arctan 15 sin &

15 cos & ' 22

M " 10&660 cos & ' 709

12.10!,

N 26.67! E,

00 90

7500

0! ≤ & < 90!T " 3000 sec &;

TBC ! 2169.51 poundsTAC ! 3611.11 pounds,

! 45.00

! 53.62

62.72!

-v- ! 647.85, & ! 44.11!

y

x−100 100 200 300 400 500

−100

100

200

300

400

500 647.85

44.11°



d 0! 1! 2! 3!

Force 0 523.57 1046.98 1570.08

d 4! 5! 6! 7!

Force 2092.69 2614.67 3135.85 3656.08

d 8! 9! 10!

Force 4175.19 4693.03 5209.45

d 0 200 400 800

Work 0 2,717,760.92 5,435,521.84 10,871,043.69

45.

47. 49.

51. 0 53. 55.

57. 32

59. (a) 37,289; It is the total dollar value of the picture framesproduced by the company.

(b) Multiply by 1.02.

61. (a) Force 30,000

(b)

(c) 29,885.84 pounds

63. (a)

(b)

65. foot-pounds 67. 725.05 foot-pounds

69. True. The zero vector is orthogonal to every vector.

71. Orthogonal.

73. (a) and are parallel. (b) and are orthogonal.

75 and 77. Proof

79. is a horizontal shift of four units to the right.

81. is a vertical shift of six units upward.

83. 85. 87. 10 89.

91. 93.


Vocabulary Check (page 456)1. absolute value

2. trigonometric form, modulus, argument

3. DeMoivre’s 4. th root

1. 3.

6 4

5. 7.

9. 11.

13. 15.

17. 19.

2'cos %6

' i sin %6(5&2'cos

7%4

' i sin 7%4 (

Realaxis

Imaginaryaxis

−1

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1

2

1 2

3 + i

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−4

−3

−2

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1

5432

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6' i sin

11%6 (2&2'cos

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' i sin 5%4 (

2"cos % ' i sin %#3'cos %2

' i sin %2(

3&54&2

54321−1−2−3−4

5

34

8

67

21

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−2

−1

1

2

3

4

1

− 4 + 4i

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−5 −4 −3 −2 −1

−3

−2

−1

1

2

3

1

− 4

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1

−1

2

3

4

5

6

7

1 2 3 4

6i

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n

54321−1−2−3−4

5

34

8

67

21

−2

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1 + 8i

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34

21

−2−3−4−5

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4726 $ 27

26i15 ' 12i$1 ' 2i

fg

fg

vuvu

u , v " 0

21,650.64

W " 15,691&32

d

sin d!"

v

$34i $ 1

2j, 34i ' 12j)3, $1*, )$3, 1*

u45229)2, 15*, u " ) 90

229, 675229* ' )$ 90

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17, 5217*


CH

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21. 23.

25.

27. 29.

31.

33.

35.

37. 39.

41. 43.

3 ' 3&3 i$15&2

8'

15&28

i

54321−1−2−3−4

5

34

8

67

21

−2

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3 + 3 3i

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−2−3− 4

15 28

15 28

− i+

1

2

3

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3&34

$34

i$1 ' &3 i

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3 3 3 i−4 4

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3

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−1 + 3i

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21

−2

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3

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3

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Realaxis

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axis



45. 47.

49. 51.

53.

The absolute value of each is 1.

55. 57.

59. 61.

63. 65.

67. (a)

(b) and (c) 4

69. (a)

(b) and (c) 4

71. (a)

(b) and (c)

73. (a)

(b) and (c)

75. (a)

(b) and (c)

77. (a)

(b) and (c)

79. (a)

(b) and (c)

81. (a)

(b) and (c)

83. 85.

87. 89.

91. 93. 95.

97. 99. 101.

103. 256 105. 107.


113. (a)

(b) and (c) 8i

2"cos 270! ' i sin 270!#2"cos 150! ' i sin 150!#;2"cos 30! ' i sin 30!#;

9&22

'9&2

2i

2048 ' 2048&3 i$597 $ 122i

4.5386 $ 15.3428ii125

2'

125&32

i

$32&3 ' 32i$32i$2 ' 2i

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θ = π6

axis

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θ = π6

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2 6−1

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54

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2&2'cos %4

' i sin %4(

5"cos 0 ' i sin 0#

3 $ 3&34

'3 ' &3

4i

2 'cos 5%3

' i sin 5%3 (

3&2'cos %4

' i sin %4(

2 $ 2i

&2 'cos 7%4

' i sin 7%4 (

2"cos 0 ' i sin 0#

$2 $ 2&3 i

2 'cos 11%

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2 'cos 3%2

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2 $ 2i

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cos 30! ' i sin 30!1150"cos 130! ' i sin 130!#

109

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CH

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115. (a)

(b) and (c) 1

117. 119.

121.

123.

125. (a)

(b)

(c)

127. (a)

(b)

(c)

129. (a)

(b)

(c)

131. (a)

(b)

(c)

133. (a)

(b)

(c) 2&3 ' 2i, $2&3 ' 2i, $4i

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135. (a)

(b)

(c)

137. (a)

(b)

(c)

139. (a)

(b)

(c)

141.

143.

145.

147.

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CH

AP

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R 6


149. 151. 153.

155. True 157. True 159. Answers will vary.


163. Maximum displacement: 16;

165. Maximum displacement:

167. 169.

Review Exercises (page 461)1.

3.

5.

7. No solution

9.

11. Two solutions

13. 9.08 15. 221.34

17. 15.29 meters 19. 31.01 feet

21.

23.

25.

27.

29.

31.

33. 4.29 feet, 12.63 feet 35. 1135.45 miles

37. 9.80 square units 39. 511.71 square units

41. 43. 45.

47. 49.

51.

53. (a) (b) (c)

(d)

55. (a) (b) (c)

(d)

57. (a) (b) (c)

(d)

59. (a) (b) (c) (d)

61. 63.

65. 67. 69.

71.

73.

Magnitude: 32.62

Direction:

75. 76.79 pounds 77. 180 pounds

79. 422.30 miles per hour, 81. 83. 7

85. 25 87. 89. 2.802 91.

93. 95.

52.2!90!

y

x

u

v

−2−4 64 8 10 12−2

−4

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x

y

−5 5 10 15 20 25 30−5

5

10

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3%4 (j%

9i $ 8j&2929

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10 20 30 40 90

−20−10

−30−40−50

1020304050

x

y

5

10

5 10 15 20 25 30

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A " 35!, C " 35!, b ! 6.55

A ! 86.38!, B ! 28.62!, c ! 22.70

A ! 13.19!, B ! 20.98!, C ! 145.83!

A ! 15.29!, B ! 20.59!, C ! 144.11!

A ! 82.82!, B ! 41.41!, C ! 55.77!

A ! 119.48!, B ! 10.52!, b ! 5.23

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3934 ' 3

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2 i34 ' 38i



97. Parallel 99. Neither 101. 103.

105.

107. 109. 72,000 foot-pounds

111. 113.

1

115.

117. 119.

121. 123.

125. (a)

(b) and (c) 12

127. (a)

(b) and (c)

129. (a)

(b) and (c)

131. 133.

135. 137.

139.

141. (a) (b)

(c)

143. (a) (b)

(c)

145.

147.

149. True 151. Direction and magnitude

153. z1z2 " $4, z1

z2" $

14

z12

2'cos 11%

6' i sin

11%

6 (2'cos

7%

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7%

6 (

−3

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3

1

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$2.1213 $ 2.1213i,$2.8978 ' 0.7765i,

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CH

AP

TE

R 6


Chapter Test (page 465)1.

2.

3.

4. Two solutions

5. No solution 6.

7. 675 feet 8. 2337 square meters

9.

10. (a) (b) (c)

11. (a) (b) (c)

12. (a) (b) (c)

13. (a) (b) (c)

14. 15.

16. 250.15 pounds 17. 3

18. 19. No, because not 0.

20.

21. 22.

23. 24.

25.

26.

Cumulative Test for Chapters 4–6(page 466)1. (a) (b)

(c)

(d)

(e) sin

cos

tan

csc

sec

cot

2. 3.

4. 5.

6. 7.

8. 9. 10. 11.


16. 17. 1.7646, 4.5186%6

' n%, 5%6

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35

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y

1

2

3

x

y

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2

3

−2

−3

2

1

3

π6

− π6

π2

−3 −2 −1−1

1

1

2

3

2 3x

y

cos & " $ 513146.1!

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"$150!# " $2&3

3

"$150!# " $2

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6

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w " )12, 13*, -w- ! &313

B ! 14.79!, C ! 15.21!, c ! 4.93

B ! 138.90!, C ! 16.10!, c ! 11.81

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A ! 22.33!, B ! 49.46!, C ! 108.21!

C " 46!, a ! 13.07, b ! 22.03



18. 19.

20. 21. 22.

23. 24–27. Answers will vary.

28. 29.

30. 31.

32. 131.71 square inches 33. 94.10 square inches

34. 35. 36. 37. 1

38.

39. 40.

41.

42.

43. 1,

44.

45. 5 feet 46.

47. 543.88 kilometers per hour

48.


Vocabulary Check (page 481)1. system, equations 2. solution

3. method, substitution 4. point, intersection

5. break-even point

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

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

5. 7.

9.

11. 13. 15. 17.

19. 21. No solution 23.

25. No real solution 27.

29. 31. 33. No real solution

35. 37. 39.

41. 43. 45.

47. 49. 51.

53. 55. No real solution

57. 59.

61. 63.

65.

192 units; $1,910,400

67.

3133 units; $10,308

00

5,000

15,000

C

R

00

400

3,500,000

C

R

!!4, !14", !1

2, 2"!0, 1", !1, 0"!0.287, 1.751"!0.25, 1.5"

!!2, 0", !2910, 21

10"!1, 2"!0, !13", !±12, 5"!2.25, 5.5"

!2.318, 2.841"!0, 1"!±1.540, 2.372"!8, 3", !3, !2"!4, !0.5"!3, 6", !!3, 0"

!52, 32"!4, 3"

!0, 0", !1, 1", !!1, !1"!!2, 0", !3, 5"!20

3 , 403 "

!1, 1"!12, 3"!5, 5"!4, 4"

!0, 2", !#3, 2 ! 3#3", !!#3, 2 " 3#3"!2, 6", !!1, 3"!2, 2"

80.28#

32.63#,

d $ 4 sin %4

t

3$cos 9%5

" i sin 9%5 %

3$cos 7%5

" i sin 7%5 %

3!cos % " i sin %"

3$cos 3%5

" i sin 3%5 %

3$cos %5

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i!12

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2


CH

AP

TE

R 7


Year Missouri Tennessee

1990 5104 4875

1994 5294 5181

1998 5483 5487

2002 5673 5793

2006 5862 6099

2010 6052 6405

Week Animated Horror

1 336 42

2 312 60

3 288 78

4 264 96

5 240 114

6 216 132

7 192 150

8 168 168

9 144 186

10 120 204

11 96 222

12 72 240

69. (a)

(b) and (c)

(d) The answers are the same.

(e) During week 8 the same number of animated andhorror films were rented.

71. (a)

(b)

780 units

73. Sales greater than $11,667

75. (a)

(b)

More invested at 6.5% means less invested at 8.5% andless interest.

(c) $5000

77. (a)

(b) From 1998 to 2010

(c) (d)

(e) During 1997 the populations of Missouri andTennessee were equal.

79. meters 81. miles

83. False. You can solve for either variable before back-sub-stituting.

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

87. (a) (b) (c)

89. 91.

93. 95.

97. Domain: All real numbers x except

Asymptotes:

99. Domain: All real numbers x except

Asymptotes:

101. Domain: All real numbers x

Asymptote:


Vocabulary Check (page 491)1. method, elimination 2. equivalent

3. consistent, inconsistent

y $ 0

x $ ±4y $ 1,

x $ ±4

x $ 6y $ 0,

x $ 6

30x ! 17y ! 18 $ 0y ! 3 $ 0

2x " 7y ! 45 $ 0) y $ x ! 32y $ x ! 4

y $ !2y $ 0y $ x " 1

N0 $ N,

8 miles & 126 meters & 9

!7.87, 5477.01"

!7.87, 5477.01"

45000 20

6500

00

25,000

20,000 0.065x + 0.085y = 1600

x + y = 20,000

) x

0.065x

"

"

y

0.085y

$

$

20,000

1,600

020,000

2,000

100,000

C

R

R $ 55.95x

C $ 35.45x " 16,000

x $ 8



1. 3.

5. Inconsistent

7. 9. 11. 13.

15. Inconsistent 17. b; One solution, consistent

19. c; One solution, consistent 21.

23. Inconsistent 25. All points on

27. 29.

31. All points on 33.

35. 37. 39.

41. 43.

Consistent; Inconsistent

45. 47.

Consistent; all pointson

49. 51.

53.

55. 57.

59. 61.

63. 65.

67. 69.

71. Plane: 550 miles per hour; wind: 50 miles per hour

73. (a)

(b) Answers will vary.

(c)

75. 9 oranges, 7 grapefruit

77. 185 movies, 125 video game cartridges

79. 81.

83. (a) and (b)

(c) (d) 41.4 bushels per acre

85. True. A linear system can have only one solution, no solution, or infinitely many solutions.

87. The solution is outside the viewing window.

89. No. Two lines can only intersect once, never, or infinitelymany times.

91. 93.

95. 97.

99. 101.

103. 105. 107. Answers will vary.ln x2

x ! 2log9

12x

43210−1−2−3−4−5−6x

72

ln 6x"5 < x < 72

42 6 8 10 12 14 16 180−2x

−10 −9 −8 −7 −6 −5

3−

x

22

"2 < x < 18x ≤ "223

u # "x ln x; v # ln xk # "4

!39,600, 398".

300

60

y # 14x ! 19

y # "2.5x ! 5.54y # 0.97x ! 2.1

A # 650, C # 525

A # 650, C # 525;

#5.00AA

!!

3.50CC

##

5087.501175

!2,000,000, 100"!80, 10"

#2x

x

!

"

2y

4y

#

#

11

"7#3x

x

!

!

12y

3y

#

#

4

24

!436 , 25

6 "!6, "3"!"5

3, "113 "!4, 1"

!"4, 5"

−12

−2

6

10

!1, "0.667"!6, 5"

4−2

−3

1

−3

−2

15

10

8x " 14y # 5!3.833, 7"

903

9

−6

−4

6

4

!5, 2"

−9

−6

9

6

−5

−6

13

6

!1, 12"!"1, 1"!9031, "67

31"!101, 96"5x " 6y " 3 # 0

!7, 1"!5, "2"6x ! 8y " 1 # 0

!" 635, 43

35"

!127 , 18

7 "!4, "1"!3, 4"!52, 34"

−4

−6 6

4x − y = 2−2x + 2y = 5

−4

−6 6

4 x + y = 0

3x + 2y = 1−3

−5 7

5x − y = 1

2x + y = 5

!1, "1"!2, 1"


CH

AP

TE

R 7



Vocabulary Check (page 505)1. row-echelon 2. ordered triple 3. Gaussian

4. independent, dependent 5. nonsquare

6. three-dimensional

7. partial fraction decomposition


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

5. 7. 9.

11.

It removed the term from Equation 2.

13.

It removed the term from Equation 2.

15. 17. 19.

21. Inconsistent 23.

25. 27.

29. 31. Inconsistent

33. Inconsistent 35.

37. 39.

41. 43.

45. 47.

49. 51.

53.

55.

57. 59.

61.

63. 65.

67. 69.

71. 73.

75.

77.

79.

81.

The vertical asymptotes are the same.

83. 85.

87. 89.

91. 93.

95. $366,666.67 at 8%, $316,666.67 at 9%, and $91,666.67 at 10%

97. in certificates of deposit

in municipal bonds

in blue-chip stocks

in growth stocks

99. 24 two-point field goals, 6 three-point field goals, 18 freethrows

101. Four touchdowns, four extra-point kicks, one field goal

103.

105. Four par-3 holes, 10 par-4 holes, four par-5 holes

107. 109. y $ x 2 ! xy $ ! 524 x2 ! 3

10 x " 416

I1 $ 1, I2 $ 2, I3 $ 1

g

$218,750 ! 1.75g

$125,000

$156,250 " 0.75g

x2 " y 2 " 6x ! 8y $ 0x2 " y 2 ! 4x $ 0

y $ x2 ! 6x " 8y $ 12x2 ! 2x

s $ !16t2 ! 32t " 500s $ !16t 2 " 144

–6 2 8 10

–8

2

8

x

y = 3x

y = 3x

y = 2x − 4

−

y = 2x − 4

−

y

–6 –4 2 8 10

–8

2

4

6

8

x

y

3x

!2

x ! 4

x " 3 "6

x ! 1"

4!x ! 1"2 "

1!x ! 1"3

2x ! 7 "17

x " 2"

1x " 1

3x

!3

x ! 3"

2!x ! 3"2

3x

!1x2 "

1x " 1

!3x

!1

x " 2"

5x ! 2

32x ! 1

!2

x " 11x

!2

2x " 1

1x

!1

x " 112$ 1

x ! 1!

1x " 1%

Ax ! 5

"B

!x ! 5"2 "C

!x ! 5"3

Ax

"Bx2 "

Cx ! 10

Ax

"B

x ! 14

!0, 0, 4", !0, 2, 2"!2, 0, 0", !0, 4, 0",

x y

4

6 644

6

z

!0, 0, 3", !4, 0, 1"!6, 0, 0", !0, 4, 0",

x y

4

22

6 64

6

z

)x

2x

!x

"

!

"

6y

2y

y

"

!

"

4z

4z

z

$

$

$

7

0

!74

)3x

x

!x

"

"

"

y

2y

y

!

!

"

z

z

3z

$

$

$

9

0

1

!1, !1, 2"!1, !2, !1"!1

2, 0, 12, 32"!!5a " 3, !a ! 5, a"!!3

2 a " 12, !2

3 a " 1, a"!2a, 21a ! 1, 8a"!!1, 1, 0"

!!a " 3, a " 1, a"!1, !1, 2"!!3a " 10, 5a ! 7, a"

!1, !32, 12"

!2, !3, !2"!!4, 8, 5"!1, 2, 3"x-

) x

3x

!

!

2y3yy

""!

zz

4z

$$$

1!2

1

x-

)x

2x

! 2y

y

"

!

!

3z

2z

3z

$

$

$

5

9

0

!12, !2, 2"!3, 10, 2"!2, !2, 2"



x !2 !1 0 1 2

y 16 8 4 2 1

x !2 !1 0 1 2

y !0.5 0 1 3 7

111. (a)

(b)

(c) 453 feet

113. (a)

(b)

115. False. The leading coefficients are not all 1.

117. False.

119. 121.

123. No. There are two arithmetic errors. The constant in thesecond equation should be and the coefficient of inthe third equation should be 2.

125.

127. 129.

131. 133.

135. (a)

(b)

137. (a)

(b)

139.

141.


−1

−2

1 2 3

12345678

x

y

−1−2 1 2 3 4 5 6 7 8

1234

89

x

y

642−2−6

20

10

−30

−40

−50

−60

x

y

!4, !32, 3

25

20

−5

−10

−15

642−2−6x

y

!4, 0, 3

65421−1−2−3−4

765

−2−3

x

y

5432−2−3−4−5

1

−5−6−7−8−9

x

y

x $ 0, y $ 0, ' $ 0

7654321−1−2−3

54321

67

x

y

x $ !#22

, y $12

, ' $ 1

x $#22

, y $12

, ' $ 1

x $ 5, y $ 5, ' $ !5

z!11

1a$1

y"

1a ! y%1

2a$ 1a " x

"1

a ! x%

Ax " 10

"B

x ! 10"

C!x ! 10"2

100

700 20007 − 4x

200011 − 7x

20007 ! 4x

!2000

11 ! 7x, 0 ≤ x ≤ 1

6000

500

y $ 0.165x2 ! 6.55x " 103

) 900a1600a2500a

"""

30b40b50b

"""

ccc

$$$

55105188


CH

AP

TE

R 7



Vocabulary Check (page 521)1. matrix 2. square

3. row matrix, column matrix 4. augmented matrix

5. coefficient matrix 6. row-equivalent

7. reduced row-echelon form

8. Gauss-Jordan elimination

1. 3. 5.

7.

9.

11. 13.

15.

17.

19. Add times to 21. Interchange and

23. Reduced row-echelon form

25. Not in row-echelon form 27. Not in row-echelon form

29. (a) (b)

(c) (d)

(e)

The matrix is in reduced row-echelon form.

31. (a) (b)

(c) (d)

(e)

The matrix is in reduced row-echelon form.

33. 35.

37. 39.

41. 43.

45. 47. 49.

51. Inconsistent 53. 55.

57. 59.

61. 63.

65. Yes; 67. No 69.

71. 73.

75.

77. $150,000 at 7%, $750,000 at 8%, $600,000 at 10%

79.

81. (a)

(b)

(c) 2005: $67.58; 2010: $86.33; 2015: $103.08(d) Answers will vary.

83.

50 pounds of glossy, 35 pounds of semi-gloss,15 pounds of matte

85. (a)

(b)

(c)

x5 $ 1000, x6 $ 0, x7 $ !500

x1 $ 0, x2 $ !500, x3 $ 600, x4 $ 500,

x6 $ 0, x7 $ 0

x1 $ 0, x2 $ 0, x3 $ 600, x4 $ 0, x5 $ 500,

x4 $ s ! t, x5 $ 500 ! t, x6 $ s, x7 $ t

x1 $ s, x2 $ t, x3 $ 600 ! s,

)5.5g!g

g

"""

4.25sss

"""

3.75mmm

$$$

4800

100

0 50

80

y $ !0.0400t 2 " 4.350t " 46.83

I1 $ 1310, I2 $ 11

5 , I3 $ 910

f !x" $ x3 ! 2x2 ! 4x " 1

f !x" $ !9x2 ! 5x " 11y $ 2x2 ! x " 1

y $ x2 " 2x " 5!!1, 1, !3"!!5a, a, 3"!0, 2 ! 4a, a"

!7, !3, 4"!2a " 1, 3a " 2, a"!4, !3, 2"!3, !2, 5, 0"

!3, 2"!!4, !8, 2"!7, !5"

!8, 0, !2"!!2, !3"

)x ! y

y

"

!

2z

z

z

$

$

$

4

2

!2

)x ! 2y

y

$

$

4

!3

*1

0

0

1

!37

!127

!87

107+*

100

010

001+

*100

!110

!160

130+*1

0

0

3

1

0

32

314

1

5

0

!3512+

*100

010

!120+

*100

210

320+*

1 0 0

2 !5

0

3!10

0+

*1 0 0

2 !5 !5

3!10!10

+* 1 0 3

2 !5

1

3!10!1

+

R2.R1R1.R2!3

*100

153

4!220

!164+, *

100

113

4!2

5

20

!165

4+

*10

42

3!1+

)9x

!2x

x

"

"

"

12y

18y

7y

"

"

!

3z

5z

8z

$

$

$

0

10

!4

)3x

x

"

!

4y

y

$

$

9

!3

*1

5

2

10

!3

1

!2

4

0

!!!

2

0

6+

* 6

!2

!7

5!!

11

!1+2 & 23 & 11 & 2



87. False. It is a matrix.

89.

(Answer is not unique.)

91. Gauss-Jordan elimination was not performed on the lastcolumn of the matrix.

93. 95.

Asymptotes: Asymptotes:


Vocabulary Check (page 536)1. equal 2. scalars 3. zero, 4. identity

5. (a) iii (b) i (c) iv (d) v (e) ii

6. (a) ii (b) iv (c) i (d) iii

1. 3.

5. (a) (b)

(c) (d)

7. (a) (b)

(c) (d)

9. (a)

(b)

(c)

(d)

11. (a) Not possible (b) Not possible

(c) (d) Not possible

13. 15.

17. 19.

21. 23.

25. Not possible 27.

29. 31.

33. (a) (b) (c)

35. (a) (b) (c)

37. (a) (b) (c) Not possible

39. 41.

43. Not possible 45. 47.



53. (a) (b)

55. (a) (b)

57. (a) (b)

59. (a) (b)

61. (a) (b)

The answers are the same.

* 7!616

!21311

5!8!3+* 7

!616

!21311

5!8!3+

*!13

!2+* 1!3

0

!51

!2

2!1

5+*x1

x2

x3+ $ *!20

8!16+

* 1!1

2+* 1!1

2

!23

!5

3!1

5+*x1

x2

x3+ $ * 9

!617+

*!76+*!2

6!3

1+*x1

x2+ $ * !4

!36+*4

8+*!1!2

11+*

x1

x2+ $ *4

0+

*!43

1014+* 5

!48

!16+

*15151647

25279

!20

4838787+*70

3216

!1711

!38

736

70+,13-*

78

!1

7 8

!1

1416

!2+

*86

!68+* 0

10 !10

0+* 0 10

!100+

*1115

614+*!3

392

14+*08

1511+

*!15!18

!5!6

!25!30

!45!54+*1

00

010

0072+

*!2!8

0

513327+

* 3

!12

!132

3

0112+*!6

!1 17

!90

!10+*!14.646 !41.546

78.117

21.306!69.137!32.064

+*!11

5!2+

*!24!12

!432

1212+*!8

15!7!1+

* 18!3

0!12

90+

*1015

15!10

!1!10

73

1214+

*123

156

!3!6

9!3

120+

*37

5!6

0!4

22

47+

* 5!5

510

!20

4!4

4!7+

* 22 8

!14

!1519

!5+* 246

!12

!39

15+* 7

3!5

!78

!5+* 91

!3

5!215+

*!1 8

!1!19+*3

6!3!3+

*!1 3

0!9+*3

1!2

7+x $ !1, y $ 4, z $ 6x $ !4, y $ 22

O

y $ x " 2x $ 4,y $ 0x $ !1,

4

−4

−8 128

12

16

8

x

y

128−4−8−12

4

−4

−12

−8

x

y

)x

x

2x

"

"

"

y

2y

y

"

"

"

7z

11z

10z

$

$

$

!1

0

!3

2 & 4


CH

AP

TE

R 7


63. (a) (b)


65. (a) (b)


67. (a) and (b)

69. Not possible, undefined 71. Not possible, undefined

73. (a) and (b)

75. 77.

79.

The entries represent the total profit made at each outlet.

81.

The entries are the total wholesale and retail prices of theinventory at each outlet.

83.

represents the changes in party affiliations after twoelections.

85. True. To add two matrices, you add corresponding entries.

87. Not possible 89. Not possible 91.



99.

101. (a)

The entries on the main diagonal are and

(b)

is the identity matrix.

103. (a)

(Answers are not unique.)

(b) and are zero matrices.

(c)

is the zero matrix.

(d) is the zero matrix.

105. 107.


Vocabulary Check (page 547)1. square 2. inverse 3. nonsingular, singular


11. 13. 15. Does not exist

17. 19. Does not exist

21. Does not exist 23.

25. 27.

29. 31. 33.

35. 37. 39. 41.

43. 45. 47.

49.

51. Not possible, because is not invertible.

53. 55.

57.

59. !5, 0, !2, 3"!0.3125t " 0.8125, 1.8175t " 0.6875, t"

!!1, 3, 2"!!4, !8"A

!2, !2"!2, 1, 0, 0"!3, 8, !11"!!8, !6"

!5, 0"k $ 35k $ 0*!1

!32

0

!12+

*5

131

13

! 3132

13+1

59* 16!4

1570+*

14

!14

18

!58+

*1020

0101

1010

0102+5

11 *0

!2222

!411

!6

211

!8+

*!12!4!8

!5!2!4

!9!4!6+

*1

!33

12

!3

!1!1

2+*!3

!221+* 1

2

0013+

ln$#x!x " 5"#x ! 8 %ln$ 64

3#x2 " 3%An

A4

A $ *0000

2000

3500

4670+

B3A2

A $ *00

20+, B $ *0

00

200

340+

B2

B2 $ *10

01+

i 4 in A4.i 3 in A3,i 2 in A2,

A4 $ *10

01+

A3 $ *!i0

0!i+,A2 $ *!1

00

!1+,

AC $ BC $ *22

33+, A ( B

2 & 3

2 & 2

P2

*0.400.280.32

0.150.530.32

0.150.170.68+

*$15,770 $26,500 $21,260

$18,300$29,250$24,150

+

,$1037.50 $1400.00 $1012.50-

*8442

60120

3084+*!4

802+*!6

6!12!6

120+

*!1!5

10!1

!40+

* 25!53!76

!343430

28!721+* 25

!53!76

!343430

28!721+

*261111

112014

0!3

0+*261111

112014

0!3

0+A232 Answers to Odd-Numbered Exercises and Tests


61. (a) (b)

(c) they correspond to the same point.

(d)

(e) represents

63. (a) (b)

(c) they correspond to the same point.

(d)

(e) represents

65. $10,000 in AAA-rated bonds, $5,000 in A-rated bonds, and $10,000 in B-rated bonds

67. $20,000 in AAA-rated bonds, $15,000 in A-rated bonds,and $30,000 in B-rated bonds

69. amperes, amperes, amperes

71. 0 muffins, 300 bones, 200 cookies

73. 100 muffins, 300 bones, 150 cookies

75. (a)

(b)

(c) 2 pounds of French vanilla, 4 pounds of hazelnut,4 pounds of Swiss chocolate

77. (a)

(b)

(c)

(d) 2005: 7,503,000 snowboarders

2010: 20,583,000 snowboarders

2015: 45,513,000 snowboarders

(e) Answers will vary.

79. True. 81. Answers will vary.

83. 85.

87. 89.



Vocabulary Check (page 556)1. determinant 2. minor 3. cofactor

4. expanding by cofactors 5. triangular

6. diagonal

1. 4 3. 16 5. 28 7. 9. 0

11. 13.

15. (a)

(b)

17. (a)

(b)

19. (a) (b) 21. (a) 170 (b) 170

23. 25. 27. 412 29.

31. 33. 35. 410

37. (a) (b) (c) (d) 6

39. (a) 2 (b) (c) (d)

41. (a) (b)

(c) (d) 5500


53. 55. 57.

59. 61. 63. 65. 1 ! ln xe5x8uv ! 1x $ 3

x $ 1, 12x $ !4, !1x $ 1 ± #2

x $ ±32x $ ±2

*!7!413

!2

!16!14

43

!1!11

42

!288

!42+

!220!25

!12* 1!1

0

402

330+!6

*!20

0!3+!2!3

!336!168

!60!168!58

!75!75

C23 $ 6, C31 $ 54, C32 $ 53, C33 $ !34

C11 $ 10, C12 $ 43, C13 $ 2, C21 $ 30, C22 $ 17,

M33 $ !34

M22 $ 17, M23 $ !6, M31 $ 54, M32 $ !53,

M11 $ 10, M12 $ !43, M13 $ 2, M21 $ !30,

C11 $ !5, C12 $ !2, C21 $ !4, C22 $ 3

M11 $ !5, M12 $ 2, M21 $ 4, M22 $ 3

!0.002!9

!24

e12.7

3( 1.851ln 3 ( 1.099

x2 " 2x ! 13x!x ! 2" , x ( ±3

92x " 6

, x ( 0

AA!1 $ I $ A!1A

0 50

8,000

y $ 237t 2 ! 939t " 6273

A!1 $ *12

!72

6

!1

6

!8

12

!52

3+

) 4a9a

16a

"""

2b3b4b

"""

ccc

$$$

534355896309

*210

2.511

31

!1+ * fhs + $ *26

100+

)2ff

""

2.5hhh

""!

3sss

$$$

26100

I3 $ 5I2 $ 8I1 $ !3

X.A!1B $ * 2!4

1+ A!1B

A!1 $ *100

010

3!4

1+B $ *!1

01+;

!!1, 0"!2, !4"

X.A!1B $ *!321+ A!1B

A!1 $ *100

010

!211+

B $ *!111+;

!!1, 1"!!3, 2"


CH

AP

TE

R 7


67. True


Sample answer:

71. (a) 6 (b) (c) (d) They are reciprocals.

73. (a) 2 (b)

(c) (d) They are reciprocals.

75. (a) Columns 2 and 3 are interchanged.

(b) Rows 1 and 3 are interchanged.

77. (a) 5 is factored from the first row.

(b) 4 and 3 are factored from the second and third columns,respectively.

79. 81. 83.


Vocabulary Check (page 567)1. collinear 2. Cramer’s Rule 3. cryptogram

4. uncoded, coded

1. 3. 5. 24 7. 9. Collinear

11. Not collinear 13. 15.

17. Not possible 19. 21.

23. (a) and (b) 25. (a) and (b)

27. (a)

(b)


29. Uncoded:

Encoded: 21 35 14 39 35 6015 32 12 27 23 0

31. 38 63 51 58 119 133 44 88 95

33. HAPPY NEW YEAR

35. YANKEES WIN WORLD SERIES

37. True. Cramer’s Rule divides by the determinant.


41. 43.

45.

Asymptote:

Review Exercises (page 570)1. 3. 5.

7. 9.

11. 13. 15.

17. 4762 units 19. meters 21.

23. 25. 27.

29. 31.

Consistent; Inconsistent

33.

Consistent;

35.

37. 218.75 miles per hour; 193.75 miles per hour

39. 41.

43. 45.

47.

49.

Sample answer: !0, 0, 8", !0, !2, 0", !4, 0, 0", !1, !1, 2"

x y

22

64

64

8

z

!a ! 4, a ! 3, a"!!19

6 , 1712, 13"!3a " 4, 2a " 5, a"

!3817, 40

17, !6317"!4, 3, !2"

!500,0007 , 159

7 "!!4.6, !8.6"

−11

−11

7

1

!1.6, !2.4"

−7

−6

11

6

−4

−6

8

2

!145 " 8

5a, a"!0, 0"!!12, 45"

!52, 3"96 meters & 144

!!1, 1"!4, 4"!!1, 2", !0.67, 2.56"!0, 0", !4, !4"!2, !0.5"

!0, 0", !2, 8", (!2, 8"!5, 4"!1, 1"

y $ 2

1

2 31−1−2−3

3

−1

−2

−3

x

y

2x ! 7y ! 27 $ 0x " 4y ! 19 $ 0

!32!14!1

!23!54!62!115!66!68

,15 13 15-, ,18 18 15-, ,23 0 0-,3 1 12-, ,12 0 13-, ,5 0 20-,

−1 50

1,500

y $ !26.36t 2 " 241.5t " 784

!1, !1, 2"!0, !12, 12"

!!1, 3, 2"!327 , 30

7 "!!3, !2"x $ 3

x $ 0, !165

338

52

!2, !4"!2y ! 3"2!x ! 2"!x ! 1"

12

*!4!1!1

!5!1!1

1.50.5

0+16*

1313

!1316+

/A " B/ $ !30, /A/ " /B/ $ !10

A $ * 1!2

34+, B $ *!4

305+



51. (a) 53.

55. 57.

59. Spray X: 10 gallons; spray Y: 5 gallons; spray Z: 12 gallons

61. 63.

65.

67.

69.

71. 73.

75. 77. 79.

81. 83. 85.

87. 89. 91.

93. 95.

97. (a) (b)

(c) (d)

99. (a) (b)

(c) (d)

101. 103.

105. 107.

109. 111.

113. 115.

117.

119. (a)

The dairy mart’s sales and profits on milk for Friday,Saturday, and Sunday

(b) $190.80


123. 125.

127. 129.

131. 133. 135.

137. 139. 141.

143. 145. 147. 550

149. (a)

(b)

151. (a)

(b)

153. 130 155. 157. 279 159.

161. 16 163. 1.75 165. 167. 48

169. Not collinear 171. 173.

175. 177.

179. (a) and (b)

181. Uncoded:

Encoded:

183. I WILL BE BACK

185. THAT IS MY FINAL ANSWER

187. (a)

(b)

(c) and (d) or 2010t ( 20.15

0 500

40

y $ 0.41t " 11.74

2, !39, 41, !2, !39

!30, !2, 24, 38, 8, !51, 3, 0, !3, 32,

,9 0 8-, ,1 22 5-, ,0 1 0-, ,4 18 5-, ,1 13 0-

!5333, !17

33, 6166"!0, !2.4, !2.6"!!1, 4, 5"

!4, 7"!1, 2"

132

!96!3

C22 $ 19, C23 $ !22, C31 $ 5, C32 $ 2, C33 $ 19

C11 $ 30, C12 $ 12, C13 $ !21, C21 $ !20,

M33 $ 19

M22 $ 19, M23 $ 22, M31 $ 5, M32 $ !2,

M11 $ 30, M12 $ !12, M13 $ !21, M21 $ 20,

C11 $ 4, C12 $ !7, C21 $ 1, C22 $ 2

M11 $ 4, M12 $ 7, M21 $ !1, M22 $ 2

!42!1, 1, !2"!!3, 1"!1, !2, 1, 0"!2, !1, !2"

!1, !25"*!1

21

20

141

40+*1

4!1!7

2+

*101

001

2!1

3+*151

10

15

! 115+

* 13!12

5

6!5

2

!43

!1+*45

!5!6+

*438.20612.10717.28

47.4066.2077.20+

* 112

1736+

*14 19 42

!22!41!66

228066+*11

18!639+

*141436

!2!10!12

84048+1

3* 9!410

2110+

*!147

!17

!4!17!2+*32

6

!172

46

!32

33+* 9

!9!7

4+*!130

!8 11

18!19+

* 6!7

9

1523

!1

10206+*24

2012

0!4

8

288

12+*691

!5!9

3

6!4

2+*615

571

884+

*3741

!57!4+* 28

!41220+

* !3!15

238+*17

13!17

2+x $ 1, y $ 11x $ 12, y $ !7

!2, 6, !10, !3"!3, 0, !4"!1, 2, 12"!2, !3, 3"!3a " 1, !a, a"!1

2, !13, 1"

!!0.2, 0.7"!10, !12"*100

010

001+

*10

01

34

!2!3+*1

00

110

111+

)5x4x9x

"""

y2y4y

"

"

7z

2z

$$$

!9103

*835

!7!5

3

42

!3

!!!

122026+

*35

!104

!!

1522+

1 & 13 & 1

y $ 2x2 " x ! 52x ! 1x2 " 1

"!1

x " 2

12$ 3

x ! 1!

x ! 3x2 " 1%3

x " 2!

4x " 4


CH

AP

TE

R 7



189. True

191. An matrix has an inverse if

Chapter Test (page 576)1. 2.

3. 4. 5.

6. 7.

8. 9.

10. 11.

12. (a) (b)

(c) (d)

13. 14. 67 15.

16. 30 17.

18.


Vocabulary Check (page 587)1. infinite sequence 2. terms 3. finite

4. recursively 5. factorial

6. summation notation

7. index, upper limit, lower limit 8. series

9. th partial sum

1. 7, 9, 11, 13, 15 3. 2, 4, 8, 16, 32

5. 7.

9. 11. 13.

15. 17.

19. 3, 15, 35, 63, 99 21. 23. 25.

27. 29.

31.

33. 9, 15, 21, 27, 33, 39, 45, 51, 57, 63

35.

37. 0, 2, 0, 2, 0, 2, 0, 2, 0, 2

39. c 40. b 41. d 42. a

43. 45. 47.

49. 51. 53.

55.

57. 28, 24, 20, 16, 12 59. 3, 4, 6, 10, 18

61. 6, 8, 10, 12, 14;

63. 81, 27, 9, 3, 1; 65.

67. 69. 71.

73. 495 75. 77. 79. 35

81. 40 83. 30 85. 87. 238 89. 30

91. 81 93. 95.

97. 99.

101. 103.

105. 107. 109. (a) (b)

111. (a) (b)

113. (a)

(b) $6741.74

A8 ! $5307.99A7 ! $5268.48,

A6 ! $5229.26,A5 ! $5190.33,

A4 ! $5151.70,A3 ! $5113.35,

A2 ! $5075.28,A1 ! $5037.50,

19

111110,000

23

33335000"3

27516

!5

i!1 2i " 12i#1 " 2.0156!

20

i!1 #"1$i#1

i2 " 0.821

!6

i!1#"1$i#13i ! "546!

8

i!1 %2& i

8' # 3( ! 33

!9

i!1 13i

" 0.942994760

95

12n#2n # 1$n # 1

112

1, 12

, 124

, 1

720,

140,320

1, 13

, 25

, 67

, 83

1, 1, 12

, 16

, 124

an !2433n

an ! 2n # 4

an ! #"1$n # 2#1$n ! #"1$n # 2

an !1n!

an ! 1 #1n

an !#"1$n#1

2n

an !n # 1n # 2

an ! n2 " 1an ! 3n " 2

3, 52, 73, 94, 115 , 13

6 , 157 , 17

8 , 199 , 21

10

00 11

3

−10

0 11

20

00 11

8

6465

100101"73

"1, 14

, "19

, 1

16, "

125

1, 1

23)2, 1

33)2, 1

43)2 !18

, 1

53)2

12, 34, 78, 15

16, 31320, 1, 0, 12, 01

2, 25, 310, 4

17, 526

2, 32, 43, 54, 65"12, 14, "1

8, 116, " 1

32

n

x5 ! tx4 ! 100 " t,x3 ! s,

x2 ! 300 " s " t,x1 ! 700 " s " t,

#1, "12$

"2%"43

"4313

"53

"8323

1

1

0(, #"2, 3, "1$

% 36"28

10

20"24

8

4"4

2(% 7"18

1

4"16

10

12"2

0(% 15"12

3

12"12

6

1200(% 1

"70

0"6

4

4"1

0(#5, 2, "6$#"2a # 1.5, 2a # 1, a$

1x

#2x2 "

1x2 # 1

5x " 1

#3

#x " 1$2

y ! "12 x2 # x # 6#1, 0, "2$

#"23, "1

2, 1$#289 , "31

9 $#8, 5$, #2, "1$#0, "1$, #1, 0$, #2, 1$#4, "2$

det#A$ $ 0.A"1An % n


115. (a)

(b) 4625; 3969; 3477

(c) 2000 trout; Answers will vary. Sample answer: Astime passes, the population of trout decreases at adecreasing rate. Because the population is growingsmaller and still declines 25%, each time 25% is takenfrom a smaller number, there is a smaller decline inthe number of trout.

117. (a)

(b) $616.78 (c) $2832.26

119. (a)

(b) 2010: $12.25; 2015: $12.64

(c) Answers will vary. (d) 2008

121. (a)

(b) Linear regression sequence:0.98656

Quadratic regression sequence:0.99919

(c) Linear regression sequence:

Quadratic regression sequence:

(d) Quadratic regression sequence; the coefficient ofdetermination for the quadratic regression sequence iscloser to 1.

(e) 2010: $912,850,000; 2015: $1,494,560,000

(f) 2010

123. True 125. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144;

127. 1, 1, 2, 3, 5

129.

131. 133.

135.

137.

139.

141.

143.

145. 0, ln 2, ln 3, ln 4, ln 5;

147. (a) (b)

(c) (d)

149. (a) (b)

(c) (d)


Vocabulary Check (page 598)1. arithmetic, common 2.

3. th partial sum

1. Arithmetic sequence,




9. 21, 34, 47, 60, 73

Arithmetic sequence,

11.

Not an arithmetic sequence

13. 143, 136, 129, 122, 115

Arithmetic sequence, d ! "7

12, 13, 14, 15, 16

d ! 13

d ! 0.6

d ! 8

d ! "12

d ! "2

n

an ! dn # c

%161013

314722

423125(%"2

41

74223

"164548(

% 8"12"3

17"13"15

"14"9

"10(%"341

"744

413(

% 027

618(%18

1890(

%"223

"7"18(% 8

"317(

ln#n!$

16

, 112

, 120

, 130

, 142

; 12

"1

n # 2

14

, 112

, 124

, 140

, 160

; 12

"1

2n # 2

x # 1, "#x # 1$2

2,

#x # 1$3

6, "

#x # 1$4

24,

#x # 1$5

120

"x, x2

2, "

x3

6,

x4

24, "

x5

120

"x2

2,

x 4

24, "

x6

720,

x8

40,320, "

x10

3,628,800

"x3

3,

x5

5, "

x7

7,

x9

9, "

x11

11x,

x2

2,

x3

6,

x 4

24,

x5

120

an#2 !32

an ##1 # *5$n

# #1 " *5$n

2n#1

an#1 !12

an ##1 # *5$n

# #1 " *5$n

2n#1

1, 2, 32, 53, 85, 138 , 21

13, 3421, 55

34, 8955

8 18100

600

8 18100

600

Rn ! 3.088n2 " 22.62n # 130.0;

Rn ! 54.58n " 336.3;

8 18100

600

8 159

12

a0 ! 50, an ! an"1#1.005$ # 50

p0 ! 5500, pn ! #0.75$pn"1 # 500


CH

AP

TE

R 8


Year 1997 1998 1999 2000

Sales (in billionsof dollars)

12.1 13.0 13.9 14.8

Year 2001 2002 2003 2004

Sales (in billionsof dollars)

15.7 16.6 17.5 18.4


15. 1, 5, 1, 5, 1

Not an arithmetic sequence

17. 19.

21. 23.

25. 27. 5, 11, 17, 23, 29

29. 31.

33. 22.45, 20.725, 19, 17.275, 15.55

35.

37.

39. 59 41. 18.6

43. 45.

47.

49. 19.25, 18.5, 17.75, 17, 16.25, 15.5, 14.75, 14, 13.25, 12.5

51. 1.55. 1.6. 1.65. 1.7. 1.75. 1.8. 1.85. 1.9. 1.95, 2

53. 110 55. 57. 2550 59.

61. 620 63. 41 65. 4000 67. 1275

69. 25,250 71. 355 73. 129,250 75. 440

77. 2575 79. 14,268 81. 405 bricks

83. $150,000

85. (a)

(b)

The model fits the data.

(c) $122.0 billion

(d) $180.3 billion; Answers will vary.

87. True. Given you know that Hence,

89. 91. 4

93. (a)

(b) 17, 23, 29, 35, 41, 47, 53, 59

(c) Not possible

(d) 4, 7.5, 11, 14.5, 18, 21.5, 25, 28.5

(e) Not possible

Answers will vary.

95. 20,100 97. 2601 99.

101. 15 square units 103. Answers will vary.


Vocabulary Check (page 607)1. geometric, common 2.

3.

4. geometric series 5.

1. Geometric sequence,

3. Not a geometric sequence



9. Not a geometric sequence 11. 6, 18, 54, 162, 486

13. 15.

17.

19. 64, 32, 16, 8, 4; 21. 9, 18, 36, 72, 144;

23.

25. 27. 29. 31.

33. 35.

37. 39.

41. 8, 4, 6, 5

00 11

24

−10

0 11

14

an ! 5#6$n"1; 50,388,480an ! 7#3$n"1; 45,927

"29

500#1.02$13"2

310

1128

an ! 6#"32$n"1

6, "9, 272 , "81

4 , 2438 ; r ! "3

2;

an ! 9#2$n"1an ! 64#12$n"1

r ! 2;r ! 12;

1, e, e2, e3, e 4

5, "12, 1

20, " 1200, 1

20001, 12, 14, 18, 116

r ! 2

r ! "12

r ! 3

S ! !&

i!0 a1r i !

a1

1 " r

Sn ! !n

i!1 a1r i"1 ! a1&1 " r n

1 " r 'an ! a1r n"1

#1, 5, "1$

a1 ! 4, an#1 ! an # 3.5

a1 ! 17, an#1 ! an # 6

a1 ! "7, an#1 ! an # 3

"7, "4, "1, 2, 5, 8, 11

x, 3x, 5x, 7x, 9x, 11x, 13x, 15x, 17x, 19x

an ! a1 # #n " 1$d.d ! a2 " a1.a1 and a2,

an ! 0.91n # 5.7

"4585"25

"1, 3, 7, 11, 15, 19, 23, 27, 31, 35

00 11

10

00 11

16

35, 12, 25, 3

10, 15; d ! " 110, an ! " 1

10 n # 710

15, 19, 23, 27, 31; d ! 4; an ! 11 # 4n

"2, 2, 6, 10, 14"10, "22, "34, "46, "58

an ! 103 " 3n

an ! 103 n # 5

3an ! 132 " 5

2n

an ! 108 " 8nan ! "2 # 3n


45. 511 47. 43

49. 29,921.31 51. 6.4 53. 2092.60

55. 57. 59. 50 61.

63. Series does not have a finite sum because

65. 67. 69. 32 71.

73. 75.

77. (a) $1343.92 (b) $1346.86 (c) $1348.35

(d) $1349.35 (e) $1349.84

79. $6480.83 81. Answers will vary.

83. (a) $26,198.27 (b) $26,263.88

85. (a) $153,237.86 (b) $153,657.02

87. 126 square inches

89. (a) (b)

(c)

3.5 hours

91. $1600 93. $1250 95. $2181.82

97. (a) Option 1. Answers will vary. Sample answer: Youmake a cumulative amount of $157,689.86 fromoption 1 and $169,131.31 from option 2.

(b) Option 2. Answers will vary. Sample answer: Youmake about $33,114.39 from option 1 and $35,179.05from option 2 the year prior to re-evaluation.

99. (a) 3208.53 feet; 2406.40 feet; 5614.93 feet total

(b) 5950 feet

101. False. A sequence is geometric if the ratios of consecutiveterms are the same.

103. 105.

107. (a)

Horizontal asymptote:

Corresponds to the sum of the series

(b)

Horizontal asymptote:

Corresponds to the sum of the series

109. Divide the second term by the first to obtain the commonratio. The term is the first term times the commonratio raised to the power.

111. 45.65 miles per hour 113.



Vocabulary Check (page 617)1. mathematical induction 2. first

3. arithmetic 4. second

1. 3.

5.

7–19. Answers will vary. 21. 1,625,625 23. 572


43. 0, 3, 6, 9, 12

First differences:

Second differences:

Linear

45.

First differences:

Second differences:

Quadratic

47.

First differences: 1, 2, 3, 4

Second differences: 1, 1, 1

Quadratic

0, 1, 3, 6, 10

"1, "1, "1

"2, "3, "4, "5

3, 1, "2, "6, "11

0, 0, 0

3, 3, 3, 3

1 # 6 # 11 # . . . # #5k " 4$ # #5k # 1$

2k#1

#k # 2$!5

#k # 1$#k # 2$

"102

#n " 1$thnth

y ! 10

−6

−6 24

14

y ! 12

−20

−3 9

28

100e8x3, 3x2

, 3x2

4,

3x3

8,

3x 4

16

0 140

75

18.4'F; 4.8'FTn ! 70#0.8$n

11390

411

94"30

19100089

+73+ > 1.

103!

7

n!12#"1

4$n"1!7

n!15#3$n"1


CH

AP

TE

R 8

43.

1 162 243 284 305 316 31.57 31.758 31.8759 31.9375

10 31.96875

Snn


49. 2, 4, 6, 8, 10



Linear

51. 53.

55. (a)

(b) (c)

57. False. Not necessarily

59. False. It has second differences.

61. 63.

65. 67.


Vocabulary Check (page 624)1. binomial coefficients

2. Binomial Theorem, Pascal’s Triangle

3. 4. expanding, binomial

1. 21 3. 1 5. 15,504 7. 14 9. 4950

11. 749,398 13. 4950 15. 31,125

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

37.

39.

41.

43.

45.

47.

49. 51. 53.

55. 57. 59. 180

61. 63. 210 65. 21 67. 6

69.

71.

73.

75.

77.

79.

81.

83. 85. 87.

89. 91.

93. 1 95. 97. 1.172 99. 510,568.785

101.

is shifted three units to the left of

103. 105. 0.273 107. 0.171

is the expansion of

109. (a)

(b)

111. False. The correct term is

113. The first and last numbers in each row are 1. Every othernumber in each row is formed by adding the two numbersimmediately above the number.

115. The terms of the expansion of alternate betweenpositive and negative.

117 and 119. Answers will vary.

121. is shifted eight units up from

123. is the reflection of in the axis.y-f #x$g#x$f #x$.g#x$

#x " y$n

126,720x4y8.

0−5 18

g

f

600

g#t$ ! 0.064t 2 " 6.74t # 256.1, "15 ≤ t ≤ 3

f #x$.p#x$

−3

−6 6

5

h

f = p

g

g#x$ ! x3 # 9x2 # 23x # 15

f.g

−6

−10 8

6

fg

"18

"23 # 208*3 i"115 # 236i

2035 # 828i161 # 240i"4

h $ 0*x # h " *x

h!

1*x # h # *x

,

6x5 # 15x 4h # 20x3h2 # 15x2h3 # 6xh4 # h5, h $ 0

3x2 # 3xh # h2, h $ 0

x2 " 3x 4)3y1)3 # 3x2)3y2)3 " y

x2 # 20x3)2 # 150x # 500x1)2 # 625

# 810xy 4 " 243y532x 5 " 240x 4y # 720x3y 2 " 1080x2y 3

81t 4 " 216t 3v # 216t 2v 2 " 96tv 3 # 16v4

"489,888

3,247,69532,476,950,000x 4y8

1,259,712x2y7360x3y261,440x7

"4x6 " 24x5 " 60x4 " 83x3 " 42x2 " 60x # 20

2x4 " 24x3 # 113x2 " 246x # 207

"512x 4 # 576x3 " 240x2 # 44x " 3

16x4 "

32yx3 #

24y2

x2 "8y 3

x# y 4

1x5 #

5yx4 #

10y2

x3 #10y3

x2 #5y4

x# y5

x18 " 6x15y # 15x12y2 " 20x9y 3 # 15x6y 4 " 6x3y 5 # y6

x8 # 4x6y2 # 6x 4y 4 # 4x2y6 # y8

x10 " 25x8 # 250x6 " 1250x 4 # 3125x2 " 3125

x8 # 8x6 # 24x 4 # 32x 2 # 16

1 " 12x # 48x2 " 64x3

x5 " 5x4y # 10x3y2 " 10x2y3 # 5xy4 " y5

# 576rs5 # 64s6

729r6 # 2916r5s # 4860r 4s2 # 4320r3s3 # 2160r2s 4

x5 # 5x4y # 10x3y2 # 10x2y3 # 5xy4 # y5

y4 " 8y3 # 24y2 " 32y # 16

a3 # 9a2 # 27a # 27

x 4 # 8x3 # 24x2 # 32x # 16

nCr or &nr'

40#1 # 3*2$7*3i

"64x3 # 240x2 " 300x # 1254x4 " 4x2 # 1

n " 2

3&43'

n"1an !

*34 %1 #

13 !

n

k!2&4

9'k"2(

an ! 3#4$n"1

an ! 12 n2 # n " 3an ! n2 " 3n # 5, n ≥ 1



n 10 15 20 23 30 40 50

Pn 0.88 0.75 0.59 0.49 0.29 0.11 0.03

Qn 0.12 0.25 0.41 0.51 0.71 0.89 0.97

125.


Vocabulary Check (page 634)1. Fundamental Counting Principle 2. permutation

3. 4. distinguishable permutations

5. combinations

1. 6 3. 5 5. 3 7. 7 9. 120 11. 1024

13. (a) 900 (b) 648 15. 16,000,000

17. (a) 35,152 (b) 3902

19. (a) 100,000 (b) 20,000 21. (a) 720 (b) 48

23. 24 25. 336 27. 120 29. 27,907,200

31. 197,149,680 33. 120 35. 362,880

37. 11,880 39. 50,653

41. ABCD, ABDC, ACBD, ACDB, ADBC, ADCB,

BACD, BADC, CABD, CADB, DABC, DACB,

BCAD, BDAC, CBAD, CDAB, DBAC, DCAB,

BCDA, BDCA, CBDA, CDBA, DBCA, DCBA

43. 420 45. 2520 47. 10 49. 4 51. 1

53. 4845 55. 850,668

57. AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE,DF, EF

59. 61. 13,983,816 63. 36

65. 3744 67. (a) 495 (b) 210

69. 292,600 71. 5 73. 20

75. or 77.

79. 81. 83. False.

85. For some calculators the answer is too large.

87. They are equal. 89 and 91. Answers will vary.

93. 8.303 95. 35 97. 99.


Vocabulary Check (page 645)1. experiment, outcomes 2. sample space

3. probability 4. impossible, certain

5. mutually exclusive 6. independent

7. complement 8. (a) iii (b) i (c) iv (d) ii

1.

3.

5.

7. 9. 11. 13. 15. 17.

19. 21. 23. 0.25 25. 27. 0.88 29.

31. (a) 1.22 million (b) 0.41 (c) 0.24 (d) 0.26

33. (a) 37.59 million (b) 0.01 (c) 0.012

35. (a) 0.45 (b) 0.45 (c) 0.23

37. (a) (b) (c)

39.

41. (a) (b) (c)

43. (a) (b) 45. (a) 0.008875 (b)

47. (a) (b)

49. (a) (b) (c) 51. (a) (b) (c)

53. 0.1024 55. (a) 0.9702 (b) 0.9998 (c) 0.0002

57. (a) (b) (c)

59. (a) (b) Answers will vary. 61. True

63. (a) As you consider successive people with distinct birth-days, the probabilities must decrease to take intoaccount the birth dates already used. Because the birthdates of people are independent events, multiply therespective probabilities of distinct birthdays.

(b)


(d) is the probability that the birthdays are not distinct,which is equivalent to at least two people having thesame birthday.

(e)

(f) 23

65. 67. 69.

71. 73. 60 75. 6,652,800

77. 15 79. 165

x ! 16e4 " 9.100

x ! ln 28 " 3.332x ! "10x ! 112

Qn

365365 ( 364

365 ( 363365 ( 362

365

)4

11,52915,625

409615,625

115,625

5455

1255

1455

413

12

513

6.84 % 10"66.84 % 10"7

38788

124

1120

49323 " 0.152225

646 " 0.348211292 " 0.016

P#,Moore wins-$ ! P#,Perez wins-$ ! 14

P#,Taylor wins-$ ! 12

5481254

5821254

6721254

720

13

25

15

1112

536

413

313

78

38

#B, E$, #C, D$, #C, E$, #D, E$-,#A, B$, #A, C$, #A, D$, #A, E$, #B, C$, #B, D$,

,ABC, ACB, BAC, BCA, CAB, CBA-#T, 1$, #T, 2$, #T, 3$, #T, 4$, #T, 5$, #T, 6$-,#H, 1$, #H, 2$, #H, 3$, #H, 4$, #H, 5$, #H, 6$,

#"1, 1$#"2, "8$

n ! 2n ! 3

n ! 10n ! 6n ! 5

4.418694268 % 1016

nPr !n!

#n " r$!

%45

"5"6(


CH

AP

TE

R 8


Review Exercises (page 651)1. 3. 5.

7. 9. 11.

13. 15. 30 17. 19. 51,005,000

21. 43,078 23.

25. 27. (a) (b)

29. (a) (b) 2

31. (a) 2512.50, 2525.06, 2537.69, 2550.38, 2563.13,2575.94, 2588.82, 2601.77

(b) $3051.99



37. 3, 7, 11, 15, 19 39. 1, 4, 7, 10, 13

41. 35, 32, 29, 26, 23;

43. 9, 16, 23, 30, 37;

45. 47. 80 49. 88

51. 25,250 53. (a) $43,000 (b) $192,500



59. 61. or

63.

65.

67.

69. 71. 127 73. 3277

75. 1301.01 77. 24.85 79. 32 81. 12

83. (a) (b) $20,168.40

85 and 87. Answers will vary. 89. 465 91. 4648

93. 5, 10, 15, 20, 25



Linear model

95. 16, 15, 14, 13, 12

First differences:


Linear model

97. 45 99. 126 101. 20 103. 70

105.

107.

109. 111. 10

113. (a) 216 (b) 108 (c) 36 115. 45

117. 239,500,800 119. 4950 121. 999,000

123. 5040 125. 3,628,800 127. 15,504

129. 131.

133. (a) 0.416 (b) 0.8 (c) 0.074 135. 0.0475

137. True.

139. (a) Each term is obtained by adding the same constant(common difference) to the preceding term.

(b) Each term is obtained by multiplying the sameconstant (common ratio) by the preceding term.

141. (a) Arithmetic. There is a constant difference betweenconsecutive terms.

(b) Geometric. Each term is a constant multiple of thepreceding term. In this case the common ratio isgreater than 1.

143. Each term of the sequence is defined using a previousterm or terms.

145. If is even, the expressions are the same. If is odd, theexpressions are negatives of each other.

Chapter Test (page 655)1. 2.

3.

4.

5. 7920 6. 7. 8.

9. 10.

11. 12. 13. 189

14. 28.80 15. 16. Answers will vary.

17.

18. 84 19. 1140 20. 72 21. 328,440

22. 23. 26,000 24. 12,650 25.

26. 27. (a) (b) (c) 28. 0.25160

1213600

14

1462

326n ! 3

16a 4 " 160a3b # 600a2b2 " 1000ab3 # 625b 4

257

!&

n!12&1

4'n"1

!12

n!1

23n # 1

an ! 4#12$n"1

an ! 5100 " 100n

an ! n2 # 12n1

n # 1

"x3

6, "

x5

120, "

x7

5040, "

x9

362,880, "

x11

39,916,800

"x, x2

2, "

x3

3,

x 4

4, "

x5

5

12, 16, 20, 24, 281, "23, 49, " 8

27, 1681

nn

#n # 2$!n!

!#n # 2$#n # 1$n!

n!! #n # 2$#n # 1$

19n ! 3

1241 # 2520i

a5 " 20a4b # 160a3b2 " 640a2b3 # 1280ab4 " 1024b5

x 4 # 20x3 # 150x2 # 500x # 625

"1, "1, "1, "1

at ! 120,000#0.7$t

an ! 100#1.05$n"1; 3306.60

an ! 16#"12$n"1

; 10.67

an ! 25#"35$n"1

25, "15, 9, "275 , 81

25; r ! "35;

an ! 120#13$n"1

120, 40, 403 , 40

9 , 4027; r ! 13;

9, "6, 4, "83, 16

99, 6, 4, 83, 1694, "1, 14, " 1

16, 164

r ! "13

r ! 2

an ! 103 " 3n; 1430

an ! 2 # 7n

d ! 7;

an ! 38 " 3n

d ! "3;

d ! 12

d ! "2

158

59

11112000!

9

k!1

kk # 1

! 7.071

!20

k!1 12k

! 1.799

20524#n # 1$#n$

1380

9, 5, 1, "3, "7an !2

2n " 1

an ! 5n"1, 12, "16, 1

24, " 1120

23, 45, 89, 16

17, 3233



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Chapter 5orensclass.weebly.com/uploads/1/2/3/5/12353620/chapter_5-8_odd_answers.pdf · 75. 77. 79. 81. 83. 85. 87. 89. 91. 93. 95. 97. 99. 101. The identity is not true when 103