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CPM Educational Program © 2012 Chapter 7: Page 1 Pre-Calculus with Trigonometry
Chapter 7: Algebra for College Mathematics Courses Lesson 7.1.1 7-1. a. See graph at right.
Increasing: x > 2 ; Decreasing: x < 2 b. As the x-values get larger, the y-values get larger. Or, the slope
of the tangent line is positive. 7-2. a. First x-value is less than the 2nd and both are in the interval [a, b]. b. First y-value is less than the 2nd. c. See graph at right. d. Yes, f (x1) < f (x2 ) . 7-3. a. x1 = 2!!!!!x2 = 3
22 < 23
4 < 8
x1 = 1.7!!!!!x2 = 1.821.7 < 21.8
3.25 < 3.48
b. g(x) is an increasing function on the interval [a, b] if, for every two points x and x + h with a ! x < x + h ! b, h > 0 , g(x) < g(x + h) .
c. 2x+h = 2h !2x . Since h > 0, 2h > 1 ; therefore 2h !2x > 2x . 7-4. a. Something like, “as x gets larger, y gets smaller.” b. If x2 > x1 , then f (x2 ) < f (x1) . c. f (x) is a decreasing function on the interval [a, b] if, for every two points 1x and 2x with
a ! x1 < x2 ! b , then f (x2 ) < f (x1) . d. Given h > 0 , 5 ! (x + h)2 = 5 ! h(2x + h) .
If !" < x < x + h < 0 , then 2x + h < 0 ; therefore !h(2x + h) > 0 and 5 ! (x + h)2 > 5 ! x2 . If 0 < x < x + h < ! , then 2x + h > 0 ; therefore !h(2x + h) < 0 and 5 ! (x + h)2 < 5 ! x2 .
7-6. A graph is concave down over an interval [a, b] if a line segment joining any two points on
the graph over that interval lies completely below the graph. 7-7. See graph at right. Increasing on (!", !1) and (1,!) , decreasing on (!1, 0) and
(0,1) ; concave up on (0,!) , concave down on (!", 0) .
x
y
a bx1 x2
f(x1)f(x2)
x
y f(x)
x
y
CPM Educational Program © 2012 Chapter 7: Page 2 Pre-Calculus with Trigonometry
7-8. Any line or odd function that passes through the origin, for example:
y = 5x,!y = x5,!y = sin x . Review and Preview 7.1.1 7-9. a. Increasing: (–∞, –3) ∪ (1, 5), Decreasing: (–3, 1) ∪ (5, ∞), Concave Up: (–1, 3),
Concave Down: (–∞, –1) ∪ (3, ∞); b. Decreasing: (–∞, ∞), Concave Up (–∞, 0), Concave down (0, ∞) 7-10. See sample graph at right. 7-11. Brittany; segments connecting any two points on the graph
are above the graph. 7-12. a. b(x) = a(x ! 2) + 5 = (x ! 2)3 ! 3(x ! 2) + 5
b(x) = (x ! 2)(x2 ! 4x + 4) ! 3x + 6 + 5b(x) = x3 ! 4x2 + 4x ! 2x2 + 8x ! 8 ! 3x + 6 + 5b(x) = x3 ! 6x2 + 9x + 3
b. c. W + 25 7-13. a. Amplitude : !7!(!1)
2 = 3
Period : 4"b = 2" !!#!!b = 12
y = 3sin 12 x( ) ! 4 !!or !!
y = 3 cos 12 (x ! " )( ) ! 4
b. Amplitude: 3!(!1)2 = 2
Period: "b = 2" !!#!!b = 2
y = 2 sin 2 x ! "4( )( ) +1!!or !!
y = !2 cos(2x) +1
x
y
x
y
x
y
CPM Educational Program © 2012 Chapter 7: Page 3 Pre-Calculus with Trigonometry
7-14. 2(x ! 3) + 3
x+1 = 72(x ! 3)(x +1) + 3 = 7(x +1)2(x2 ! 2x ! 3) + 3 = 7x + 7
2x2 ! 4x ! 3 = 7x + 72x2 !11x !10 = 0
7-15. tan!1 2≈ 1.107 radians, (63.435˚) or 2.034 radians (180º – 63.435º = 116.565˚). 7-16.
slope = m = s!ur!t
distance = (r ! t)2 + (s ! u)2
distance = r ! t (r!t )2 +(s!u)2
(r!t )2
d = r ! t (r!t )2
(r!t )2+ (s!u)2
(r!t )2
d = r ! t m2 +1 Lesson 7.1.2 7-17. a. b. Changing of sign of x does not affect f(x). c. y = (a)2 = a2
y = (!a)2 = a2 They are equal.
d. f (a) = f (!a) e. Symmetric about the y-axis. 7-18. a. These functions have even power exponents. b. f (!x) = f (x) c. They are symmetric about the y-axis. d. y = cos x or y = x are good choices.
x
y
(t, u)
(r, s) y = mx + b
y = (2)2 = 4y = (!2)2 = 4y = (3)2 = 9y = (!3)2 = 9y = (1.721)2 = 2.962y = (!1.721)2 = 2.962
x = !(!11)± (!11)2 !4(2)(!10)2(2)
x = 11± 2014
CPM Educational Program © 2012 Chapter 7: Page 4 Pre-Calculus with Trigonometry
7-19. a. b. Changing the sign of x changes only the sign of f (x) .
c. y = (a)3 = a3
y = (!a)3 = !a3
d. f (!a) = ! f (a) e. Symmetric around the origin. 7-20. a. These functions have odd exponents. b. f (!x) = ! f (x) c. They are symmetric about the origin. d. y = sin x is a good choice. 7-21. a. (–2, 5) b. (3, –5) c. unknown 7-23. a. Even g(!x) = (!x)2 + cos2(!x)
g(!x) = x2 + cos2 x
b. Neither f (!x) = (!x)2 + 3(!x)3
f (!x) = x2 ! 3x3
c. Odd h(!x) = (!x)!1 + 2 sin(!x)h(!x) = !x!1 ! 2 sin x
Review and Preview 7.1.2 7-24. a. cos(!x) = cos(x) and sin(!x) = ! sin x b. Sine is odd, cosine is even. 7-25. a. Any parabola with a vertex on the y-axis. Example: f (x) = x2 ! 3 b. Impossible c. Any parabola with a vertex not on the y-axis. Example: f (x) = 2x2 ! 8x + 5 7-26. Tangent is odd. tan(!x) = sin(!x)
cos(!x) =! sin xcos x = ! sin x
cos x = ! tan x 7-27. Increasing: (–2, 4); Decreasing: (–∞, –2) ∪ (4, ∞);
Concave Up: (–∞, 1); Concave Down: (1, ∞)
y = (2)3 = 8y = (!2)3 = !8y = (3)3 = 27y = (!3)3 = !27y = (1.721)3 = 5.097y = (!1.721)3 = !5.097
CPM Educational Program © 2012 Chapter 7: Page 5 Pre-Calculus with Trigonometry
7-28. a. 2.9 – 1.7 = 1.2 seconds b. 46 – 20 = 26 c. y = 13 cos 2!
1.2 (x "1.7)( ) + 33 (other answers are possible) d. Because the period of the graph is less than the horizontal shift, one solution is x = 2.0445 !1.2 = 0.8445 . The graph will have another maximum at x = 1.7 !1.2 = 0.5 . Due to the symmetry of the graph, the second solution is x = 0.5 ! (0.8445 ! 0.5) = 0.1555 . 7-29.
logbNPM 2 = 1
2 logb N + 12 logb P ! 2 logb M
= 12 "0.6 +
12 " !1.8 ! 2 "2.1
= 0.3! 0.9 ! 4.2 = !4.8
7-30. a. b. c. 7-31. Interval length = 4!1
10 = 310 = 0.3 x0 = 1, x1 = 1.3, x2 = 1.6, x3 = 1.9, x4 = 2.2, x5 = 2.5,
x6 = 2.8, x7 = 3.1, x8 = 3.4, x9 = 3.7, x10 = 4.0
30 = 13 cos 2!1.2 (x "1.7)( ) + 33
"3 = 13 cos 2!1.2 (x "1.7)( )
" 313 = cos
2!1.2 (x "1.7)( )
cos"1 " 313( ) = 2!
1.2 (x "1.7)
1.804 = 2!1.2 (x "1.7)
0.3445 = x "1.72.0445 = x
!2 f (x)
f (!x) +1
1f (x)
CPM Educational Program © 2012 Chapter 7: Page 6 Pre-Calculus with Trigonometry
0.3 40.3k+1
k=0
9
! = 6.023 . This is an upper bound because the rectangles are above the curve.
CPM Educational Program © 2012 Chapter 7: Page 7 Pre-Calculus with Trigonometry
Lesson 7.2.1 7-32. a. See sketch at right. b. h = height, x = width c. V = 4500 = 2x ! x !h = 2x2h
S = 2x ! x + 2x !h + 2x !h + x !h + x !hS = 2x2 + 6xh
d. We want to know the smallest surface area represented by the variable, S. e. 4500 = 2x2h!!!!!h = 4500
2x2= 2250
x2
S = 2x2 + 6x " 2250x2
S = 2x2 + 13500x
f. x = 15, S = 1350
h = 2250x2
= 2250225 = 10
7-33. a. s
6 =(s+20)10
10s = 6s +1204s = 120s = 30
b. s6 =
(s+4t )10
10s = 6s + 24t4s = 24ts = 6t feet
7-34. V = ! r2d = 16!d
16!d = 3t
d(t) = 3t16!
7-35.
rd = 3
88r = 3d
r = 3d8
V = 2t = ! r2d3 = ! (3d 8)2 d
3
2t = ! (3d 8)2 d3
2t = ! 9d2 "d3"64
384t = 9!d3384t9! = d3
d(t) = 4 2t3!
3
10
20 s
6
h
2x x
r
d
CPM Educational Program © 2012 Chapter 7: Page 8 Pre-Calculus with Trigonometry
Review and Preview 7.2.1 7-36. Graph or average the x-intercepts (x = 0 and x = 60) to find that the product will be a maximum when x = 30. 30 + y = 60!!!!!y = 30
30 + 30 = 60, product = 900
7-37. S = 2x2 + 2x2 + 6xh = 4x2 + 6xh
4500 = 2x2h
7-38. x2 + x2 = (x + 2)2
2x2 = x2 + 4x + 40 = x2 ! 4x ! 4
x = !(!4)± (!4)2 !4(1)(!4)2(1)
x = 4± 16+162 = 4±4 2
2 = 2 ± 2 2
Area = 12 ! (2 + 2 2) ! (2 + 2 2)
Area = 1+ 2( ) 2 + 2 2( )Area = 2 + 2 2 + 2 2 + 4
Area = 6 + 4 2
7-39. a. (7 ! 5 cos")2 = (7 ! 5 cos")(7 ! 5 cos")
= 49 ! 35 cos" ! 35 cos" + 25 cos2 "= 49 ! 70 cos" + 25 cos2 "
b. (sin! + cos!)2 = (sin! + cos!)(sin! + cos!)= sin2 ! + sin! cos! + sin! cos! + cos2 != 1+ 2 sin! cos!= 1+ sin 2!
7-40. y + y + x + x + x = 300
2y + 3x = 3002y = 300 ! 3x
y = 300!3x2
A = xy
A = x 300!3x2( )
A = 150x ! 32 x
2 This is a maximum when x = 50.
x = 502y + 3(50) = 300
2y = 150y = 75 They should be 37.5 ft wide and 50 ft long.
y
y
x x x
x + y = 60!!!!!P = xy
y = 60 ! x!!!!!P = x(60 ! x) = !x2 + 60x
x
x x+2
Diagram for 7-38.
CPM Educational Program © 2012 Chapter 7: Page 9 Pre-Calculus with Trigonometry
7-41. a. 5(x!2) = 5x "5!2 = 1
52"5x = 1
25 "5x
b. 91 2 x+1 = 32(1 2 x+1) = 3x+2 = 3x ! 32 = 9 ! 3x
c. 60 23( )2x!2 = 60 2
3( )2x 23( )!2 = 60 " 3222 " 2
3( )2#$
%&x= 60 " 94 "
49( )x = 135 4
9( )x
7-42. a. b. 7-43. sin(2A) cos(2A) = 1
4
2 sin(2A) cos(2A) = 12
sin(2 !2A) = 12
sin(4A) = 12
4A = "6 + 2"n,
5"6 + 2"n
A = "24 +
"n2 ,
5"24 +
"n2
CPM Educational Program © 2012 Chapter 7: Page 10 Pre-Calculus with Trigonometry
Lesson 7.2.2 7-44. a. u = x2 + 2 , 3u ! u = 5 b. u = 4! " 2 or u = sin(4! " 2) , sin2 u ! sin u +1 = 0 or u2 ! u +1 = 0 c. u = 3x
x2 +3, log u + 2u = 7
7-45. a. u = y!5 2
2x + u = 63x ! 2u = !5
u = 6 ! 2x
3x ! 2(6 ! 2x) = !53x !12 + 4x = !5
7x = 7x = 1
u = 6 ! 2(1) = 44 = y!5 2
y = 4!2 5
b. u = x2 + 3xu2y = 5!!!!!u = 10y
3u " 3y = 27
3(10y) ! 3y = 2727y = 27y = 1u = 10(1) = 10
(10)2 = x2 + 3x( )2100 = x2 + 3x0 = x2 + 3x !100
x = !3± 32 !4(1)(!100)2(1)
x = !3± 4092
7-46. a. u + u ! 6 = 0 b. v2 + v ! 6 = 0 c. (v + 3)(v ! 2) = 0 v ! 2 = 0 or v + 3 = 0
v = 2 v = !3
d. !3 " (M 2 + 3M !1)12 No value associated with v = –3.
(2)2 = (M 2 + 3M !1)1 2( )24 = M 2 + 3M !10 = M 2 + 3M ! 5
M = !3± 32 !4(1)(!5)2(1)
M = !3± 292
7-47. Joey needs 16 unit squares. 7-48.
CPM Educational Program © 2012 Chapter 7: Page 11 Pre-Calculus with Trigonometry
a. y = x2 + 8xy = x2 + 8x + (16 !16)y = (x2 + 8x +16) !16y = (x + 4)2 !16
b. y = x2 + 6x !1y = x2 + 6x + (9 ! 9) !1y = (x2 + 6x + 9) ! (9 +1)y = (x + 3)2 !10
CPM Educational Program © 2012 Chapter 7: Page 12 Pre-Calculus with Trigonometry
7-49. a. 9 b. c. V = (!3, !6) 7-50.
x2 + 6x + 9 + y2 ! 4y + 4 ! (9 + 4) = 51(x + 3)2 + (y ! 2)2 !13 = 51
(x + 3)2 + (y ! 2)2 = 64 center: ( 3, 2)! ; radius: 8 Review and Preview 7.2.2 7-51. a. 27 b. y = 3x2 !18x + 27 ! 27 +1
y = 3(x2 ! 6x + 9) ! 26y = 3(x ! 3)2 ! 26V = (3, !26)
7-52. y = x2 ! 8x +1
y = x2 ! 8x +16 !16 +1y = (x ! 4)2 !15V = (4, !15)
7-53. a. Let u = x2 + x !1
u2 ! 2u ! 8 = 0u ! 4( ) u + 2( ) = 0u = !2 or 4
!2 = x2 + x ! 2
x2 + x = 0x x +1( ) = 0x = 0 or !1
4 = x2 + x ! 2
x2 + x ! 6 = 0x + 3( ) x ! 2( ) = 0x = !3 or 2
! x = "3, "1, 0, 2
b. Let u = x2 + 5
12u + u = 7! u2 " 7u +12 = 0
u " 3( ) u " 4( ) = 0u = 3 or 4
x2 + 5 = 3 or x2 + 5 = 4
x2 + 5 = 9! x = ±2
x2 + 5 = 16! x = ± 11
" x = ±2 or ± 11
y ! 3 = x2 + 6x + 9 ! 9y ! 3 = (x + 3)2 ! 9
y = (x + 3)2 ! 6
CPM Educational Program © 2012 Chapter 7: Page 13 Pre-Calculus with Trigonometry
7-54. 2x + y = 120 ! y = 120 " 2x
A = xyA = x(120 " 2x)A = "2x2 +120x
x = 30y = 120 ! 2(30)= 120 ! 60 = 6060ft x 30ft
This is a max when x = 30. 7-55. a. (1! cos")2 + (sin")2 =
1! 2 cos" + cos2 " + sin2 " =1! 2 cos" +1 =2 ! 2 cos"
b. (2 sin!)2 + (2 cos!)2 =4 sin2 ! + 4 cos2 ! =
4(sin2 ! + cos2 !) =4(1) = 4
7-56. a. 4x + 3 ! 12
4x ! 9
x ! 94
b. !6 < 12 x +1 < 8
!7 < 12 x < 7
!14 < x < 14
7-57. y = k
x+6
1 = k1+6
k = 7
y = 7x+6
f (!3) = 7!3+6 =
73
f (0) = 70+6 =
76
f 13( ) = 7
1 3+6 =7
1 3+18 3 =719 3 = 7 "
319 =
2119
f 1a( ) = 7
1 a+6 =7
1 a+6a a =7
(1+6a) a = 7 "a
1+6a =7a1+6a
Lesson 7.2.3 7-58. a. b. 2x4 ! 6x3 + x3 ! 3x2 ! 2x2 + 6x + x ! 3
2x4 ! 5x3 ! 5x2 + 7x ! 3 2x4 + x3 + 4x3 + 2x2 ! 6x ! 3
2x4 + 5x3 + 2x2 ! 6x ! 3
Solution continues on next page. →
B A R N
x
y
x
2x3 +x2 !2x + 1
x 2x4 x3 –2x2 x
–3 –6x3 –3x2 6x –3
x3 +2x2 + 0x –3
2x 2x4 4x3 0x2 –6x
+ 1 x3 2x2 0x –3
CPM Educational Program © 2012 Chapter 7: Page 14 Pre-Calculus with Trigonometry
7-58. Solution continued from previous page. c. d. x4 + 2x3 ! 3x3 ! 6x2 + 2x2 + 4x + 4x + 8
x4 ! x3 ! 4x2 + 8x + 8 4x3 + 6x2 ! 2x2 ! 3x ! 4x ! 6
4x3 + 4x2 ! 7x ! 6
7-59.
2x !1 6x3 ! 5x2 + 5x ! 26x3 ! 3x2
! 2x2 + 5x! 2x2 + x
4x ! 24x ! 2
0
3x2 ! x + 2
7-60.
x +1 x5 + 0x4 ! 3x3 + 0x2 + 2x ! 5x5 + x4
! x4 ! 3x3
! x4 ! x3
! 2x3 + 0x2
!2x3 ! 2x2
2x2 + 2x !!!!!!!2x2 + 2x !!!!!!! ! 5
x4 ! x3 ! 2x2 + 2x ! 5x+1
7-61. a. b. c. The graph in Y1 follows the graph in Y2 except that it has an asymptote at x = 2. d. The quotient tells you about the general or global shape of the graph.
x3 –3x2 2x 4
x x4 –3x3 2x2 4x
+2 2x3 –6x2 4x 8
2x2 –x –2
2x 4x3 –2x2 –4x
+3 6x2 –3x –6
x ! 2 x3 ! 3x2 + 0x + 5x3 ! 2x2
! x2 + 0x! x2 + 2x
! 2x + 5!2x + 4
1
x2 ! x ! 2 + 1x!2
CPM Educational Program © 2012 Chapter 7: Page 15 Pre-Calculus with Trigonometry
e. The remainder shows where the asymptotes occur.
CPM Educational Program © 2012 Chapter 7: Page 16 Pre-Calculus with Trigonometry
Review and Preview 7.2.3 7-62. 7-63.
x !1 x5 + 0x4 + 0x3 + 0x2 + 0x !1x5 ! x4
x4 + 0x3
x4 ! x3
x3 + 0x2
x3 ! x2
x2 + 0x x2 ! x x !1 !x !1 0
x4 + x3 + x2 + x +1 x ! 3 x3 + x2 !14x + 2
x3 ! 3x2
4x2 !14x4x2 !12x
! 2x + 2!2x + 6
! 4
x2 + 4x ! 2 ! 4 x ! 3
7-64. Error is in the following line: y + 5 = !2(x2 + 2x) The line should be: y + 5 = !2(x2 ! 2x) 7-65. a. y = 2x2 ! 8x + 7
y = 2(x2 ! 4x + 4) + 7 ! 8y = 2(x ! 2)2 !1
b. Vertex is at(2, !1).
7-66. Using a graphing calculator, calculate the maximum value (ytr). The maximum value of 108 occurs when x = 3. 7-67. a. (1.02)x = 2
x log1.02 (1.02) = log1.02 2
x = log 2log 1.02 = 35
35 years
b. (1.05)x = 2x log1.05 (1.05) = log1.05 2
x = log 2log 1.05 = 14.207
14 years
c. (1.07)x = 2x log1.07 (1.07) = log1.07 2
x = log 2log 1.07 = 10.245
10 years
d. (1.1)x = 2x log1.1(1.1) = log1.1 2
x = log 2log 1.1 = 7.273
7 years
x + y = 9! y = 9 " xP = xy2
P(x) = x(9 " x)2
CPM Educational Program © 2012 Chapter 7: Page 17 Pre-Calculus with Trigonometry
e. “The Rule of 70” is called as such because the number of years to double is close to 70 divided by the annual percent growth rate.
7-68. a. x2 ! 7x " !6
x2 ! 7x + 6 " 0(x ! 6)(x !1) " 0[1, 6]!!or !1 " x " 6
b. (x ! 2)(x !1)(x + 3) < 0(!", !3)# (1, 2),! " < x < !3 !or !1 < x < 2
7-69.
a. ( x2 + y2 )3
2 x2 + y2= (x2 + y2 )3 2
2(x2 + y2 )1 2
= (x2 + y2 )3 2!1 2
2= x2 + y2
2
b. 2x5 ! 8x3
x + 2= 2x
3(x2 ! 4)x + 2
= 2x3(x + 2)(x ! 2)(x + 2)
= 2x3(x ! 2)
Lesson 7.2.4 7-70. (x + y)1 = x + y
(x + y)2 = x2 + xy + xy + y2 = x2 + 2xy + y2
(x + y)3 = (x + y)(x2 + 2xy + y2 )= x3 + 2x2y + xy2 + x2y + 2xy2 + y3
= x3 + 3x2y + 3xy2 + y3
(x + y)4 = (x + y)(x3 + 3x2y + 3xy2 + y3)= x4 + 3x3y + 3x2y2 + xy3 + x3y + 3x2y2 + 3xy3 + y4
= x4 + 4x3y + 6x2y2 + 4xy3 + y4
7-71. a. Decrease by 1 each time. b. Increase by 1 each time. c. Each time the sum is the same as the exponent of expansion. 7-73. (x + y)0 = 1 It goes in “Row 0.” 7-74. a. Row 9 b. x9 + 9x8y c. x6 + 6x5y +15x4y2 + 20x3y3 +15x2y4 + 6xy5 + y6
CPM Educational Program © 2012 Chapter 7: Page 18 Pre-Calculus with Trigonometry
d. 1, 8, 28, 56, 70, 56, 28, 8, 1
CPM Educational Program © 2012 Chapter 7: Page 19 Pre-Calculus with Trigonometry
7-75.
(x + y)15 = 150
!"
#$ x
15 + 151
!"
#$ x
14y + 152
!"
#$ x
13y2 + 153
!"
#$ x
12y3
= x15 +15x14y +105x13y2 + 455x12y3
7-76. a. x3 + 3x2y + 3xy2 + y3 b. y = 2z
x3 + 3x2(2z) + 3x(2z)2 + (2z)3
x3 + 6x2z +12xz2 + 8z3
c. (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
(x + 3w)4 = x4 + 4x3(3w) + 6x2(3w)2 + 4x(3w)3 + (3w)4
(x + 3w)4 = x4 +12x3w + 54x2w2 +108xw3 + 81w4
7-77. a. (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 b. y = !3w c. (x ! 3w)4 = x4 + 4x3(!3w) + 6x2(!3w)2 + 4x(!3w)3 + (!3w)4 d. (x ! 3w)4 = x4 !12x3w + 54x2w2 !108xw3 + 81w4 e. The signs alternate in the expansion. Review and Preview 7.2.4 7-78. x6 + 6x6!1y0+1 +15x6!2y1+1 + 20x6!3y2+1 +15x6!4y3+1 + 6x6!5y4+1 + y5+1 =
x6 + 6x5y +15x4y2 + 20x3y3 +15x2y4 + 6xy5 + y6
7-79. a. a3 + 3a2b + 3ab2 + b3 b. (2x)3 + 3(2x)2 (!3y) + 3(2x)(!3y)2 + (!3y)3 c. 8x3 ! 36x2y + 54xy2 ! 27y3 7-80. a. x2 + y2 = r2
(!6)2 + (!8)2 = r2
36 + 64 = r2
100 = r2
r = 10
b. (x ! 7)2 + (y ! 5)2 = r2
(3! 7)2 + (!2 ! 5)2 = r2
16 + 49 = r2
65 = r2
(x ! 7)2 + (y ! 5)2 = 65
CPM Educational Program © 2012 Chapter 7: Page 20 Pre-Calculus with Trigonometry
7-81. a. x2 !10x + y2 + 8y + 5 = 0
x2 !10x + 25 + y2 + 8y +16 = !5 + 25 +16(x ! 5)2 + (y + 4)2 = 36C = (5, !4), r = 6
b. x2 ! 8x + y2 + 6y ! 56 " 0x2 ! 8x +16 + y2 + 6y + 9 " 56 +16 + 9(x ! 4)2 + (y + 3)2 " 81C = (4, !3), r = 9
7-82. 7-83.
x !1 2x4 + 0x3 ! x2 + 3x + 52x4 ! 2x3
2x3 ! x2
2x3 ! 2x2
x2 + 3xx2 ! x
4x + 5 4x ! 4 9
2x3 + 2x2 + x + 4 + 9x!1
2x +1 2x5 + x4 ! 2x3 + 7x2 + 5x ! 22x5 + x4
!! 2x3 + 7x2
!!!!!!!2x3 ! x2
8x2 + 5x 8x2 + 4x
x ! 2
!!!!!! x + 12
!!! ! 52
x4 ! x2 + 4x + 12 ! 5/2
2x+1
7-84. Amplitude: 85!372 = 24 Period: 365b = 2! !!" b = 2!
365 Possible equations: y = !24 cos 2"
365 (x !17)( ) + 61
y = 24 sin 2"365 (x !107)( ) + 61
y = !24 cos 2"365 (44 !17)( ) + 61
y = !24 cos(0.4648) + 61
y = !24 "0.8939 + 61y = !21.45 + 61 = 39.5°
7-85.
a. x2 = x2( )2 + h2
h = x2 ! x24 = 3x2
4 = 32 x
b. V (x) = base ! length
V (x) = 12 xh2 = 6x 3
2 x( ) = 3 3x2
c. 200 = 3 3x2
x2 = 200 39
h = 5.373 ft
CPM Educational Program © 2012 Chapter 7: Page 21 Pre-Calculus with Trigonometry
Lesson 7.3.1 7-86. a. Each pair equals 101. b. 50 pairs c. 101 x 50 = 5050 7-87. a. 12 x 4 = 48 b. It is twice as large as A. c. 1000 by 999 7-88. a. 1100 by 899 b. 1100!899
2 = 494, 450 7-89. n = number of terms, a
1= first term of the sequence, and an = nth term of the sequence.
7-90. a. 10.2 !10 = 0.2 b. 49 times c. 49 !0.2 = 9.8
n50 = 19.825 pairs, each pair = 19.8 +10 = 29.825 !29.8 = 745
7-91.
57!298 = 3.5!!!!!S = n a1+an( )
2 = 965!!!!!a1 = 29 ! 4(3.5) = 15!!!!!an = 15 + 3.5(n !1)
965 =n 15+ 15+3.5 n!1( )( )( )
2 = n 30+3.5n!3.5( )2 = n 26.5+3.5n( )
2 = 26.5n+3.5n22
1930 = 26.5n + 3.5n2
3.5n2 + 26.5n !1930 = 0
Review and Preview 7.3.1 7-92. a. (–∞, –1) ∪ (1, ∞) b. (0, ∞) c. (–∞, 0)
d. It is odd. f (!x) = (!x)2 +1!x = ! x2 +1
x = ! f (x) 7-93. SA = 2! rh + ! r2 = 200!!"!!2! rh = 200 # ! r2 !!or !!h = 200#! r2
2! r
V (r) = ! r2h = ! r2 200"! r22! r( ) = r
2 (200 " ! r2 ) = 100r " ! r3
2
Using a graphing calculator yields a maximum value of 307.106 cm3 when r = 4.607.
If r = 4.607 then h = 200!" (4.607)22" (4.607) = 4.607 .
n = !26.5± 26.52 !4(3.5)(!1930)2(3.5) = !26.5±166.5
7 = 20
CPM Educational Program © 2012 Chapter 7: Page 22 Pre-Calculus with Trigonometry
7-94.
x + 4 2x3 + x2 !19x + 362x3 + 8x2
! 7x2 !19x!7x2 ! 28x
9x + 36 9x + 36
0
2x2 ! 7x + 9
7-95. a. 30(3+90)
2 = 1395 b. 41(20+100)2 = 2460
c. 41(20+100)2 = 2460 d. 46(37+262)
2 = 6877 7-96. a. x3 ! xy2 = x(x2 ! y2 ) = x(x + y)(x ! y) b. x3 + xy2 = x(x2 + y2 ) c. 4x2(x2 + y2 )1 2 ! 4(x2 + y2 )3 2
(x2 + y2 )1 2(4x2 ! 4(x2 + y2 ))!4y2(x2 + y2 )1 2
7-97. (1+ 2 sin!)2 + (2 cos!)2 =
1+ 4 sin! + 4 sin2 ! + 4 cos2 ! =
1+ 4 sin! + 4(sin2 ! + cos2 !) =1+ 4 sin! + 4 =5 + 4 sin!
7-98.
a. limx!4
f (x) = 4 b. limx!0"
f (x) = "2 c. limx!"#
f (x) = "3
d. limx!0
f (x) " does not exist e. limx!"3
f (x) = #
7-99. a = 200,!r = 1.01,!n = 12
200(1.0112 !1)0.01 = $2536.50
CPM Educational Program © 2012 Chapter 7: Page 23 Pre-Calculus with Trigonometry
Lesson 7.3.2 7-100. a. 1+ 3+ 9 + 27 + 81+ 243+ 729 = 1093 b. 3 ! 729 = 2187 , 3 is the multiplier. c. 3S = 3 + 9 + 27 + 81 + 243 + 729 + 2187 d. 2S = 2186 S = 1 + 3 + 9 + 27 + 81 + 243 + 729 S = 1093 3S – S = 2S = 3279 – 1093 = 2186 This is twice as much as what was found in part (a). 7-101. a. S = 1+ 5 + 25 +125 + 625 + 3125 +15625
5S = 5 + 25 +125 + 625 + 3125 +15625 + 781255S ! S = 78124 !1 = 781234S = 78123S = 19531
b.
S = 1+ 6 + 36 +!+ 77766S = 6 + 36 + 216 +!+ 466566S ! S = 46656 !1 = 466555S = 46655S = 9331
c.
S = 5 +15 + 45 +…+ 885735S = 5 ! 30 + 5 ! 31 + 5 ! 32 +…+ 5 ! 311
3S = 5 ! 31 + 5 ! 32 + 5 ! 33 +…+ 5 ! 312
3S " S = 5 ! 312 " 5 ! 302S = 2657205 " 5 = 2657200S = 1, 328, 600
d.
S = 10000 +1000 +…+ 0.001S = 104 +103 +…+10!3
10S = 105 +104 +…+10!2
10S ! S = 105 !10!39S = 100000 ! 0.0019S = 99, 999.999S = 11,111.111
7-103. The first term has no power of r, so we need to stop at (n !1) . 7-104. a. 200(1+ 0.01)11 = 200(1.01)11 b. 200(1.01)
11 + 200(1.01)10 +!+ 200 c. a = 200,!r = 1.01,!n = 12
200(1.0112 !1)0.01 = $2536.50
7-105. a. a = 200,!r = 1.01,!n = 24
200(1.0124 !1)0.01 = $5394.69
b. a = 200,!r = 1.01,!n = 60200(1.0160 !1)
0.01 = $16, 333.93
c. a = 200,!r = 1.01,!n = 120200(1.01120 !1)
0.01 = $46, 007.74
7-106. a. a = 200,!r = 1.01,!n = 240
200(1.01240 !1)0.01 = $197, 851
b. a = 200,!r = 1.01,!n = 360200(1.01360 !1)
0.01 = $698, 992.83
CPM Educational Program © 2012 Chapter 7: Page 24 Pre-Calculus with Trigonometry
c. a = 200,!r = 1.01,!n = 480200(1.01480 !1)
0.01 = $2, 352, 954.50
CPM Educational Program © 2012 Chapter 7: Page 25 Pre-Calculus with Trigonometry
Review and Preview 7.3.2 7-107. S = 1! 3+ 9 ! 27 + 81! 243+ 729 ! 2187 = !1640 . The method still works when r < 0. 7-108. a. 4(5+x)
2 = 2004(5 + x) = 4005 + x = 100
x = 9595!53 = 30
Series = 5 + 35 + 65 + 95
b. 200 = 5 + 5r + 5r2 + 5r3
195 = 5r3 + 5r2 + 5r39 = r3 + r2 + rr = 3
Series = 5 +15 + 45 +135
7-109.
a. S = 3(211!1)2!1 = 3(2047) = 6141
b. 20(800+1560)2 = 23, 600
c. S = 0.02(311!34 )3!1 = 0.02(177,066)
2 = 3541.322 = 1770.66
d. 12 +
22 +
32 +
42…+ 20
2 =20 1
2+10!
"#$%&
2 =20 21
2( )2
= 10(21)2
= 105
7-110.
S = 100(1.00512 !1)1.005!1 = 100(0.0617)
0.005 = 1233.56 7-111.
2x ! 3 4x4 ! 2x3 + 0x2 ! 7x ! 54x4 ! 6x3
!!4x3 ! 0x2
!!!! 4x3 ! 6x2
!!!6x2 ! 7x!!!6x2 ! 9x
!!!!!2x ! 5 !!!!!!!2x ! 3 !!!! ! 2
2x3 + 2x2 + 3x +1! 22x!3
7-112. 500 = 4x + 2y!!!!!y = 250 " 2x A = xy = x(250 ! 2x) This is a maximum when x = 62.5 ft. Therefore y = 125 ft and the maximum area is 7812.5 ft2.
x
y
Diagram for 7-112.
di
CPM Educational Program © 2012 Chapter 7: Page 26 Pre-Calculus with Trigonometry
7-113. a. sin A = 4
5 b. cos B = 1213
c. sin(A + B) = sin A cos B + sin B cos A= 4
5( ) 1213( ) + 5
13( ) 35( ) = 48
65 +1565 =
6365
d. cos(A + B) = cos A cos B ! sin A sin B= 3
5( ) 1213( ) ! 4
5( ) 513( ) = 36
65 !2065 =
1665
e. tan(A + B) = sin(A+B)cos(A+B) =
63/6516/65 =
6365( ) 65
16( ) = 6316
7-114. a. d = (2 ! 0)2 + (5 ! (!3))2 = 4 + 64 = 68 = 2 17
b. midpoint = 0+22 , !3+52( ) = (1,1)
slope = 5!(!3)2!0 = 8
2 = 4
slope "= ! 14
(y !1) = ! 14 (x !1)
y = ! 14 (x !1) +1
7-115. a. ar ! a = 24 !!!!!!!!!!a(r !1) = 24
ar4 ! ar3 = 648!!!!!!ar3(r !1) = 648 Dividing the equations yields:
r = 3!!!!!a(3"1) = 24 !!!!!a = 12 b. Since we are looking at the difference the equations in part (a) can be written as: ar ! a = 24
ar4 ! ar3 = 648 Other solutions will come from: ar ! a = !24
ar4 ! ar3 = !648
In this case a = –12. If r = –3, then a(!3!1) = 24 !!"!!a = !6!!!or!!!!a(!3!1) = !24 !!"!!a = 6 .
Thus all four solutions in the form (a, r) are (12, 3), (–12, 3), (–6, 3), (6, 3).
A 3
4 5
B
5
12
13
1r3
= 24648 =
127 !!!!!r = 3
CPM Educational Program © 2012 Chapter 7: Page 27 Pre-Calculus with Trigonometry
Lesson 7.3.3 PROBLEM SET A
1. 103
!"
#$
12( )3 1
2( )10%3 = 120 12( )10 & 0.117
2. 54
!"
#$ (0.6)
4 (0.4)5%4 = 5(0.6)4 (0.4)1 & 0.259
3. 44
!"
#$ (0.8)
4 (0.2)4%4 = (0.8)4 & 0.410
4. 43
!"
#$
34( )4%3 1
4( )3 = 4 34( )1 1
4( )3 & 0.0469
5. 42
!"
#$
15( )2 4
5( )2 = 6 15( )2 4
5( )2 % 0.154
6. 63
!"
#$
16( )3 5
6( )6%3 = 20 16( )3 5
6( )3 & 0.054
7. 1816
!"
#$
910( )16 1
10( )18%16 = 153 910( )16 1
10( )2 & 0.284
8. 52
!"
#$ 0.3( )2 0.7( )5%2 = 10 0.3( )2 0.7( )3 & 0.309
7-117.
a. P(R, R) = 23( ) 2
3( ) = 23( )2
b. Tracing along the branches of the tree: R, then B = 23 !13 .
c. Tracing along the branches of the tree: B, then R = 13 !23 .
d. Using the results from parts (b) and (c) indicates that the probability of getting one red and one blue = 23 !
13 +
13 !23 = 2
13 !23( ) .
7-118. a. P(B, B) = 14 !
14 =
116
b. You can get red then blue or blue then red. 7-119. a. See diagram at right. b. Using the diagram, P(2 reds) = p2 .
c. Using the diagram, P(2 blues) = q2 . d. Using the diagram, P(one red and one blue) = pq + pq = 2pq . e. p2 + 2pq + q2 = (p + q)2 = 12 = 1
p
p
q
q p
q
= p2
= pq
= pq
= q2
CPM Educational Program © 2012 Chapter 7: Page 28 Pre-Calculus with Trigonometry
7-120. a. sin2 u ! sin u + 0.24 = 0 b. Let v = sin u . c. v2 ! v + 0.24 = 0 d. v = sin(3x ! 5) 7-121. (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 Let x = a and y = bc.
a4 + 4a3(bc) + 6a2(bc)2 + 4a(bc)3 + (bc)4 =a4 + 4a3bc + 6a2b2c2 + 4ab3c3 + b4c4
7-122. a. See diagram at right. b. 0.73 = 0.343 c. 0.33 = 0.027 d. 3(0.7)2(0.3) = 0.441 e. 3(0.7)(0.3)2 = 0.189 7-123. a. y = k x
11 = k ! 411 = 2k
k = 112
y = 112 x
b. f (x) = 112 x
f (0) = 112 0 = 0
f (4) = 112 4 = 11
f (8) = 112 8 = 11
2 !2 2 = 11 2
f (a2 ) = 112 a2 = 11 a
2
7-124.
sin!1+ cos!
"1# cos!1# cos!
= sin!(1# cos!)1# cos2 !
= sin!(1# cos!)sin2 !
= 1# cos!sin!
1! cos"sin"
= 1sin"
! cos"sin"
= csc" ! cot" or tan "2( )
7-125.
0.7 0.7
0.3 0.3
0.3
0.3
0.3 0.3
0.3
0.7
0.7 0.7
0.7
0.7
CPM Educational Program © 2012 Chapter 7: Page 29 Pre-Calculus with Trigonometry
a. x2 + x ! 6 = 0(!3)2 + (!3) ! 6 = 0
9 ! 3! 6 = 0
x2 + x ! 6 = 0(2)2 + (2) ! 6 = 0
4 + 2 ! 6 = 0
b. x2 + 2x ! 6 = 0
(!1+ 7)2 + 2(!1+ 7) ! 6 = 0
1! 2 7 + 7 ! 2 + 2 7 ! 6 = 0
!2 7 + 2 7 +1+ 7 ! 2 ! 6 = 00 = 0
CPM Educational Program © 2012 Chapter 7: Page 30 Pre-Calculus with Trigonometry
7-126. d1 = d2 !!!!!t1 + t2 = 10!!!!!t2 = 10 " t1
r1t1 = r2t210t1 = 15(10 " t1)10t1 = 150 "15t125t1 = 150t1 = 6d = r1t1 = 10 #6 = 60 miles
7-127. If the y-axis is a line of symmetry then there is not a horizontal shift. The line y = 15
touches either the top or the bottom of the graph. Since the point (20, 50) is on the graph, the line y = 15 must touch the bottom. If (20, 50) is the next point of symmetry and in the middle, then the period is 80. Therefore the amplitude is 50 !15 = 35 and 80b = 2! !or !b = !
40 . Hence a possible equation is y = 35 cos ! x40( ) + 50 .
7-128. a. The zeros are at x = 1, 3, and 5. x < 1 or 3 < x < 5 The intervals to check are (!",1),!(1, 3), (3, 5),!and!(5,") . Choose a point in each interval and check to see if it makes the inequality true. (!",1)!!choose!x = 0!!#!!(1! 0)(0 ! 3)(0 ! 5) = 15 > 0!!true
(1, 3)!!choose!x = 2!!#!!(1! 2)(2 ! 3)(2 ! 5) = !3 /> 0!!false(3, 5)!!choose!x = 4 !!#!!(1! 4)(4 ! 3)(4 ! 5) = 3 > 0!!true(5,")!!choose!x = 6!!#!!(1! 6)(6 ! 3)(6 ! 5) = !15 /> 0!!false
Therefore the solution set is x < 1!!or !!3 < x < 5 . b. x2 ! 2x !15 < 0!!"!!(x ! 5)(x + 3) < 0 The zeros are at x = –3 and 5. The intervals to check are (!", !3), (!3, 5),!and!(5,") . Choose a point in each interval and check to see if it makes the inequality true. (!", !3)!!choose!x = !4 !!#!!(!4 ! 5)(!4 + 3) = 9 /< 0!!false
(!3, 5)!!choose!x = 0!!#!!(0 ! 5)(0 + 3) = !15 < 0!!true(5,")!!choose!x = 6!!#!!(6 ! 5)(6 + 3) = 9 /< 0!!false
Therefore the solution set is 3 < x < 5 . 7-129. a. d = kf
2 = k !10k = 0.2d = 0.2 f
b. d = 0.2 f
3 = 15 f
f = 15 pounds
CPM Educational Program © 2012 Chapter 7: Page 31 Pre-Calculus with Trigonometry
Chapter 7 Closure 7-130. a. The function must be cosine because it is even. If the increasing regions repeat every 4
units, then the period is 8 units. Since amplitude = 10 and 8b = 2! !or !b = !4 , a possible
equation is y = 10 cos !4 x( ) .
b. y = 12 x
odd!# c. The given information indicates that there is a vertical asymptote at x = 2 and a horizontal
asymptote at y = –1. Therefore this is a rational function. A possible equation is y = 1
x!2 !1 . d. A function that has only one horizontal asymptote is an exponential function. Since it is
concave down it is reflected over the x-axis. Since the horizontal asymptote is y = –4, the function has been shifted down 4 units. Thus a possible equation is y = !(2)x ! 4 .
e. Since the asymptotes are at x = –2 and x = 2, a possible equation is y = 1(x+2)(x!2) =
1x2 !4
.
f. This will be an odd power function centered at x = 4. A possible equation is y = !(x ! 4)3 . CL 7-131. See graph below right. a. !2 < x < 2 b. x < !2 and x > 2 c. x < 0 d. x > 0 CL 7-132. SA = 2! r2 + 2! rh
V = ! r2h
h = V! r2
SA = 2! r2 + 2! r V! r2( )
SA = 2! r2 + 2Vr
CL 7-133. 15, 050 = 100(2+a2 )
230,100 = 100(2 + a2 )301 = 2 + a2299 = a2
CL 7-134. 100 + 90 + 80 + ...! 20 !18 !16 ! ...
10(100+10)2 ! 10(20+2)2 = 1100
2 ! 2202 = 550 !110 = 440
Separate the positive terms from the negative. Each forms an arithmetic sequence.
CPM Educational Program © 2012 Chapter 7: Page 32 Pre-Calculus with Trigonometry
CL 7-135.
x ! 2 3x2 ! 2x +13x2 ! 6x
4x +1 4x ! 8
9
3x + 4 + 9x!2
The graph looks like the line 3 4x + globally, but has an asymptote at 2x = . CL 7-136. (3+ 2x !1)2 + 24 = 10(3+ 2x !1) Let u = 3+ 2x !1 . u2 + 24 = 10u
u2 !10u + 24 = 0(u ! 6)(u ! 4) = 0
u = 63+ 2x !1 = 6
2x !1 = 32x !1 = 92x = 10x = 5
u = 43+ 2x !1 = 4
2x !1 = 12x !1 = 12x = 2x = 1
u = 6!or !u = 4 Both answers check. CL 7-137. The signs are alternating so this must be subtraction. By looking at the pattern on the
exponents, the powers of x are decreasing by 1 and the power of y are increasing by 2. Therefore start with (ax ! by2 )? . The missing power must be 7 because x5(y2 )2 !!!!!5 + 2 = 7 . Now use the binomial formula to find a and b. 75
!"
#$ (ax)
5 (by2 )2 = 84x5y4
21a5x5b2y4 = 84x5y4a5b2 = 4
74
!"
#$ (ax)
4 (by2 )3 = %280x4y6
%35a4x4b3y6 = %280x4y6a4b3 = 8
Dividing the 2 new equations yields: ab =12 !!or !!2a = b .
Substitute: a5b2 = 4a5(2a)2 = 44a7 = 4a = 1!!!!!b = 2
Therefore the binomial is (x ! 2y2 )7 .
CPM Educational Program © 2012 Chapter 7: Page 33 Pre-Calculus with Trigonometry
CL 7-138. a. The series is arithmetic with a difference of 0.5. There are 161 terms in the series. S161 =
161(10+90)2 = 8050
b. The series is geometric with a ratio of 1.05. There are 21 terms in the series.
an = 20(1.05)n!120(1.05)20 = 20(1.05)n!1
20 = n !121 = n
S21 =20(1!1.0521)1!1.05 = 714.385
c. The series is geometric with a ratio of ! 12 .
There are 7 terms in the series.
S7 =400 1! ! 12( )7"
#$%
&'
1! ! 12( ) = 268.75
7-139. After 1 hour the area of the base = 81!
59 = 12
base radius (x)
5x = 108x = 21.6
After 3 hours the area of the base = 1465.74 ft2
CL 7-140. x2 ! 4x + y2 + 6y = 12
x2 ! 4x + 4 + y2 + 6y + 9 = 12 + 4 + 9(x ! 2)2 + (y + 3)3 = 25
Center (2, –3) CL 7-141. a. 5 ! 3(x+2) = k ! 3x
5!3x32
3x= k
5 !9 = kk = 45
b. 6 !2(x+k) = 24 !2x
6!2x2k
2x= 24
6 !2k = 242k = 4k = 2
c. 20 !23x"1 = 10 ! kx
2 !23x"1 = kx
2(3x"1)+1 = kx
23x = kx
23( )x = kx8x = kx
k = 8
an = 10 + 0.5(n !1)90 = 10 + 0.5(n !1)80 = 0.5(n !1)160 = n !1161 = n
an = 400 ! 12( )n!1
6.25 = 400 ! 12( )n!1
164 = ! 1
2( )n!112( )6 = ! 1
2( )n!16 = n !17 = n
slope = 1!(!3)5!2 = 4
3
" slope = ! 34
y !1 = ! 34 (x ! 5)!!or !!y + 3 = ! 3
4 (x ! 2)
CPM Educational Program © 2012 Chapter 7: Page 34 Pre-Calculus with Trigonometry
Note: x ! 0
CPM Educational Program © 2012 Chapter 7: Page 35 Pre-Calculus with Trigonometry
CL 7-142. a. 1+cos(2x)
sin(2x) =
1+1!2 sin2 x2 sin x cos x =
2!2 sin2 x2 sin x cos x =
1!sin2 xsin x cos x =
cos2 xsin x cos x =
cos xsin x = cot x
b. cos2 x4( ) ! sin2 x
4( ) =cos 2 x
4( )( ) = cos x2( )
c. (1+ cot2 y)(cos 2y +1) =
1+ cos2 ysin2 y
!"#
$%& cos2 y ' sin2 y +1( ) =
cos2 y ' sin2 y +1+ cos4 ysin2 y
' cos2 y + cos2 ysin2 y
=
1' sin2 y + cos4 y+cos2 ysin2 y
=
cos2 y + cos4 y+cos2 ysin2 y
=
cos2 y sin2 y+cos4 y+cos2 ysin2 y
=
cos2 y(sin2 y+cos2 y+1)sin2 y
=
cos2 y(2)sin2 y
= 2 cot2 y