Chapter 8, continued

Chapter 8,
7.
continued
a
d
a
10 5 }2
1
x 1 5 x2 1 x 1 1
x15
10. }
+ (x 2 1 x 1 1) 5 }
+}
3
1
x3 2 1
x 21
I 5 }2
10 5 a
(x 1 5)(x 2 1 x 1 1)
x15
5 }}
5}
2
x21
(x 2 1)(x 1 x 1 1)
For d 5 15 and a 5 10,
x2 2 2x
4x
x 2 2 6x 1 8
4x
}+}
11. } 4 }
5
2
5x 2 20
5x 2 20
x 2 6x 1 8
x 2 2 2x
a
d
10
I 5 }2
15
10
I5}
225
I 5 }2
(x 2 4)(x 2 2)
x(x 2 2)
4x
5}
+ }}
5(x 2 4)
4x(x 2 4)(x 2 2)
4
5 }}
5 }5
5x(x 2 4)(x 2 2)
I ø 0.04
If you are 15 meters from the stage, the intensity of the
sound you hear is about 0.04 watts per square meter.
2x 2 1 3x 2 5
2x 2 1 3x 2 5
1
12. } 4 (2x 2 1 5x) 5 } + }
6x
6x
2x 2 1 5x
(2x 1 5)(x 2 1)
(2x 1 5)(x 2 1)
5 437.5 1 500
5 937.5
The motorcycle is worth about $938 eight years after it
was purchased.
x21
6x
5}
5 }}
2
(6x)(x)(2x 1 5)
3500
M(8) 5 }
1 500
8
1
x(2x 1 5)
5 }}
+}
6x
3500
8. M(t ) 5 }
t 1 500
8.4 Exercises (pp. 577–580)
Skill Practice
1. To divide one rational expression by another, multiply the
first rational expression by the reciprocal of the second
rational expression.
2. A rational expression is simplified when its numerator
Lesson 8.4
8.4 Guided Practice (pp. 574–577)
2(x 1 1)
2(x 1 1)
2
1. }} 5 }} 5 }
x13
(x 1 1)(x 1 3)
(x 1 1)(x 1 3)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
(x 2 1 x 1 1)
x15
(x 2 1)(x 1 x 1 1)
5 }}
+}
2
1
and denominator have no common factors
(other than 61).
(x 2 7)(x 2 2)
x22
x 2 2 9x 1 14
3. B; }
5 }}
5}
x12
(x 2 7)(x 1 2)
x 2 2 5x 2 14
20(2x 1 1)
2 + 10 + (2x 1 1)
2(2x 1 1)
40x 1 20
2. } 5 } 5 }} 5 }
x13
10(x 1 3)
10 + (x 1 3)
10x 1 30
(x 1 2)(x 2 2)
x22
x2 2 4
4. A; }
5 }}
5}
x17
(x 1 7)(x 1 2)
x 2 1 9x 1 14
4
3. The expression } cannot be simplified.
x(x 1 2)
(x 1 7)(x 2 2)
x17
x 2 1 5x 2 14
5. C; }
5 }}
5}
x22
(x 2 2)(x 2 2)
x 2 2 4x 1 4
x14
1
x14
4. }
5 }}
5}
x24
(x 1 4)(x 2 4)
x 2 2 16
(x 2 3)(x 1 1)
x11
x 2 2 2x 2 3
5. }
5 }}
5}
x12
(x 2 3)(x 1 2)
x2 2 x 2 6
4x + x
x
4x 2
6. }
5}
5}
5x 2 3
4x(5x 2 3)
20x 2 2 12x
(x 2 5)(x 1 4)
x 2 2 x 2 20
7. }
5 }}
(x 1 5)(x 2 3)
x 2 1 2x 2 15
2x(x 1 5)
2x
2x 2 1 10x
6. }}
5 }}
5}
3x 1 1
(3x 1 1)(x 1 5)
3x 2 1 16x 1 5
7. New tin: S 5 2(2s)2 1 4(2s)(h) 5 2(4s2) 1 8sh
5 8s2 1 8sh
V 5 (2s)2(h) 5 4s2h
4s(2s 1 2h)
8s 2 1 8sh
2s 1 2h
4s(sh)
sh
4s h
18x 6y 4
2 + 9 + x4 + x2 + y2 + y2
x 2y 2
3x 5y 2 6xy 2
8. } + }
5}
5 }}
5}
4
8xy
72x 4y 2
2 + 9 + 4 + x4 + y 2
9x3y
S
V
}5}
5}5}
2
2x(x 2 5)
2x 2 10x x 1 3
x13
9. }
+}
5 }}
+}
(x 1 5)(x 2 5) 2x + x
x 2 2 25
2x 2
2
2x(x 2 5)(x 1 3)
x13
5 }}
5}
(2x)(x)(x 1 5)(x 2 5)
x(x 1 5)
The expression cannot be simplified.
(x 1 6)(x 2 4)
x24
x2 1 2x 2 24
8. }
5 }}
5}
x11
(x 1 6)(x 1 1)
x 2 1 7x 1 6
(x 2 8)(x 2 3)
x23
x 2 2 11x 1 24
9. }}
5 }}
5}
x15
(x 2 8)(x 1 5)
x 2 2 3x 2 40
(x 1 2)(x 1 2)
x 2 1 4x 1 4
10. }
5 }}
(x 2 4)(x 2 1)
x 2 2 5x 1 4
The expression cannot be simplified.
2(x2 1 x 2 2)
2(x 1 2)(x 2 1)
2(x 2 1)
2x 2 1 2x 2 4
11. }
5 }}
5 }}
5}
x27
(x 1 2)(x 2 7)
x 2 2 5x 2 14
x 2 2 5x 2 14
x24
1
x24
12. }
5 }}
5}
x 2 1 4x 1 16
x 3 2 64
(x 2 4)(x 2 1 4x 1 16)
(x 1 6)(x 2 6)
x26
x 2 2 36
13. }}
5 }}
5}
x16
(x 1 6)(x 1 6)
x 2 1 12x 1 36
3x(x2 1 2x 1 4)
3x
3x 3 1 6x2 1 12x
14. }}
5 }}
5}
3
x22
x 28
(x 2 2)(x2 1 2x 1 4)
Algebra 2
Worked-Out Solution Key
433
Chapter 8,
continued
(4x 2 1)(2x 1 3)
4x 2 1
8x 2 1 10x 2 3
15. }}
5 }}
5}
3x 1 2
(3x 1 2)(2x 1 3)
6x 2 1 13x 1 6
(5x 2 2)(x 1 4)
5x 2 1 18x 2 8
16. }}
5 }}
(5x 1 2)(2x 2 1)
10x 2 2 x 2 2
x2 1 3x 2 4
2x 2 1 4x
30. }
+}
x 2 1 4x 1 4 x 2 2 4x 1 3
(x 1 4)(x 2 1)
2x(x 1 2)
5 }}
+ }}
(x 1 2)(x 1 2) (x 2 3)(x 2 1)
2x(x 1 4)(x 2 1)(x 1 2)
The expression cannot be simplified.
x 2 2 3x 2 10
31. }
+ (x2 1 10x 1 21)
x 2 2 2x 2 15
18. Variable terms that are not factors were factored out.
x2 1 16x 2 80
x 2 16
(x 1 20)(x 2 4)
(x 1 4)(x 2 4)
x 1 20
x14
}}
5 }} 5 }
2
19. You can only divide out common factors. Since the
factors that are divided out are not common factors of
the entire numerator and denominator, you cannot divide
them out.
x 2 1 16x 1 48
x 1 8x 1 16
(x 1 12)(x 1 4)
(x 1 4)(x 1 4)
x 1 12
x14
}}
5 }} 5 }
2
x 2 2 3x 2 10 x 2 1 10x 1 21
x 2 2x 2 15
(x 2 5)(x 1 2)(x 1 7)(x 1 3)
5 }}}
5 (x 1 2)(x 1 7)
(x 2 5)(x 1 3)
5}
+ }}
2
1
x 2 1 5x 2 36
32. }
+ (x 2 2 11x 1 28)
x 2 2 49
x2 1 5x 2 36 x 2 2 11x 1 28
x 2 49
(x 1 9)(x 2 4)(x 2 4)(x 2 7)
(x 1 9)(x 2 4)2
}}
5 }}}
5
x17
(x 1 7)(x 2 7)
+ }}
5}
2
1
4x 2 1 20x x 2 1 8x 1 16
4x 2 1 20x
33. }
+ (x2 1 8x 1 16) 5 }
+}
1
x 3 1 4x 2
x3 1 4x2
4x(x 1 5)(x 1 4)(x 1 4)
5 }}
x + x(x 1 4)
20. B; The numerator and denominator factor
(x 1 4)(x 1 2)
as }}. The numerator and denominator
(x 1 3)(x 2 1)
have no common factors (other than 61), so the
4(x 1 5)(x 1 4)
x 2 1 6x 1 8
x 1 2x 2 3
5 }}
x
expression }
is in simplified form.
2
30xy4
5x2y3
y3
5x 2y 3
}
}
}4
34. }
4
5
+
7
3
x
y
30xy
x7
5x2y6
5 + x2 + y4 + y2
y2
}}
}6
5}
8 4 5
2
6
4 5
30x y
5+6+x +x +y
6x
21. P 5 4(2x) 5 8x
A 5 (2x)(2x) 5 4x 2
P
A
8x
4x
4+2+x
4+x+x
2
x
} 5 }2 5 } 5 }
10xy
8x 2y 2z x4z
8x 6y 2z 2
8x 2y2z
}
}
}
35. }
4}
4 5
3 + 10xy 5
3
xz
xz
10x 2yz 3
xz
2
4
2 + 4 + x + x + y + y + z2
4x4y
5 }}
5}
2
2
5z
2+5+x +y+z +z
(x 1 3)(x 2 2)
(x 1 3)(x 2 2)
x13
x
36. }} 4 } 5 }} + }
x
x(x 1 1)
x13
x(x 1 1)
22. P 5 x 1 (3x 2 x) 1 x 1 x 1 x 1 x 1 3x
5 x 1 2x 1 7x
5 10x
A 5 x(3x) 1 x(x) 5 3x 2 1 x 2 5 4x 2
P
A
10x
4x
5+2+x
2+2+x+x
5
2x
} 5 }2 5 } 5 }
x(x 1 3)(x 2 2)
23. The field in Exercise 21 because the perimeter of the
field in Exercise 21 is smaller and the areas of both fields
are the same.
5 }}
x(x 1 4)
16x(x 2 4)
5}
x14
x2 2 14x 1 45
x 2 2 6x 2 27
38. }
4 }}
2
x2
2x 1 2x
5 (x 2 3)(x 1 3)
2(2)(x)(x 1 5)(x 1 1)
2(x 1 1)
4(x 1 5) x(x 1 1)
27. }
+ } 5 }}
5}
x
2(x)(x)(x 1 5)
2(x 1 5)
x2
3(x 2 4)
3x 2 12 x 1 6
x16
28. } + } 5 } + }
x15
x15
2x 2 8
2(x 2 4)
3(x 1 6)
5 }}
5}
2(x 1 5)(x 2 4)
2(x 1 5)
2(x2 2 16)
x15
x15
2x2 2 32
29. } + }
5}
+ }}
4(x 2 4) (x 1 5)(x 2 5)
4x 2 16 x2 2 25
2(x 1 5)(x 1 4)(x 2 4)
x14
5 }}
5}
2(2)(x 2 4)(x 1 5)(x 2 5)
2(x 2 5)
434
Algebra 2
Worked-Out Solution Key
2(x 2 4)
8x 2
x
8x 2
37. } 4 } 5 } + }
x14
2(x 2 4)
x14
x
16(x)(x)(x 2 4)
5x 3y 4
5 + x3 + y2 + y2
y2
5x 3y
y3
}
}}
}
}
24. }
+
5
5
5
4
2
3
2
3x
15x y
5+3+x +x+y
x 2y 2 15x 2
48x7y4
8 + 6 + x3 + x4 + y4
48x5y 3
x 2y
8x4
25. }
+}
5}
5 }}
5}
6x 3y 6
6 + x3 + y4 + y2
y2
y4
6x3y 2
x(x 2 3)(x 1 3)(x 2 2)
x(x 2 3) (x 1 3)(x 2 2)
26. } + }}
5 }}
x
x(x 2 2)
x22
3(x 2 4)(x 1 6)
x22
5 }}
5}
x11
x(x 1 1)(x 1 3)
x 2 2 6x 2 27
x2
2x 1 2x
x 2 14x 1 45
(x 2 9)(x 1 3)
x2
5 }}
+ }}
2x(x 1 1)
(x 2 9)(x 2 5)
5}
+ }}
2
2
x + x + (x 2 9)(x 1 3)
x(x 1 3)
5 }}
5 }}
2x(x 1 1)(x 2 9)(x 2 5)
2(x 1 1)(x 2 5)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
2
2x(x 1 4)
5 }}}
5 }}
(x 1 2)(x 1 2)(x 2 3)(x 2 1)
(x 1 2)(x 2 3)
2
x (x 2 5) 2 3(x 2 5)
x 2 5x 2 3x 1 15
17. }}
5 }}
(x 2 5)(x 2 3)
x 2 2 8x 1 15
2
(x 2 3)(x 2 5)
x2 2 3
5 }}
5}
x23
(x 2 5)(x 2 3)
3
Chapter 8,
continued
x 2 2 4x 2 5
x 2 2 4x 2 5
1
39. } 4 (x 2 1 6x 1 5) 5 } + }
x15
x15
x2 1 6x 1 5
(x 2 5)(x 1 1)
5 }}
(x 1 5)(x 1 5)(x 1 1)
x25
(x 1 5)
5 }2
4x 1 16
3x 2 1 13x 1 4
3x 2 1 13x 1 4
x12
40. }}
4}
5 }}
+}
2
x
1
2
4x 1 16
x2 2 4
x 24
(3x 1 1)(x 1 4)
x12
5 }}
+}
(x 1 2)(x 2 2) 4(x 1 4)
(x 1 6)(x 2 6)
x 2 2 36
45. y 5 } 5 }} 5 x 1 6
x26
x26
x 2 2 36
is the same as the graph of
The graph of y 5 }
x26
y 5 x 1 6 except that there is a hole at (6, 12).
x2 2 36
y5}
x26
y (6, 12)
(3x 1 1)(x 1 4)(x 1 2)
5 }}
4(x 1 2)(x 2 2)(x 1 4)
2
3x 1 1
5}
4(x 2 2)
x22
x2 2 x 2 2
x 2 2 x 2 2 5x 1 25
41. }
4}
5}
+}
2
5x
1
25
x22
x 1 4x 2 5
x 2 1 4x 2 5
(x 2 2)(x 1 1) 5(x 1 5)
5 }}
+}
(x 1 5)(x 2 1)
x22
2
x
(2x 2 5)(x 1 2)
2x2 2 x 2 10
46. y 5 } 5 }} 5 2x 2 5
x12
x12
2x 2 2 x 2 10
}
The graph of y 5
is the same as the graph
x12
5(x 2 2)(x 1 1)(x 1 5)
of y 5 2x 2 5 except that there is a hole at (22, 29).
5(x 1 1)
y5}
x12
5 }}
(x 1 5)(x 2 1)(x 2 2)
5}
x21
2x 2 2 x 2 10
2
x 2 2 8x 1 15
42. }
4 (x 2 2 x 2 20)
x2 1 4x
y
x
21
2
x 2 8x 1 15
1
x 1 4x
x 2 x 2 20
(x 2 5)(x 2 3)
x23
5 }}
5 }2
x(x 1 4)(x 2 5)(x 1 4)
x(x 1 4)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
5}
+}
2
2
x 2 1 12x 1 32 x 2 2 49
x 2 1 4x
x 2 1 12x 1 32
43. }} 4 }
5 }}
+}
2
6x 1 42
6x 1 42
x 2 49
x 2 1 4x
(x 1 8)(x 1 4) (x 1 7)(x 2 7)
5 }}
+ }}
6(x 1 7)
x(x 1 4)
(22, 29)
47. Let a and b be the unknown side lengths of the triangle.
(x 1 8)(x 2 7)
5 }}
6x
(x 1 7)(x 1 3)
x 2 1 10x 1 21
44. y 5 }} 5 }} 5 x 1 7
x13
x13
2
x 1 10x 1 21
is the same as the graph
The graph of y 5 }}
x13
of y 5 x 1 7 except that there is a hole at (23, 4).
6x
15x
2
2
a 5 (6x) 1 (8x)2
2
2
a 5 36x 1 64x
b2 5 (15x)2 1 (8x)2
2
b2 5 225x 2 1 64x 2
a2 5 100x2
b2 5 289x 2
a 5 10x
b 5 17x
P 5 10x 1 17x 1 6x 1 15x 5 48x
1
1
1
A 5 }2 (6x 1 15x)(8x) 5 }2 (21x)(8x) 5 }2 (168x 2) 5 84x 2
x 2 1 10x 1 21
y 5 }}
x13
P
A
48x
84x
4 + 12 + x
7 + 12 + x + x
4
7x
} 5 }2 5 } 5 }
y
Problem Solving
(23, 4)
2
22
b
a 8x
(x 1 8)(x 1 4)(x 1 7)(x 2 7)
5 }}}
6x(x 1 7)(x 1 4)
x
1
1
1
48. Vp 5 }Bh 5 } (2r)2h 5 } (4)r 2h
3
3
3
1
Vc 5 }3 :r 2h
1
}(4)r 2h
Vp
3
4
}5}5}
:
1
Vc
}:r 2h
3
Algebra 2
Worked-Out Solution Key
435
Chapter 8,
Total
Average
Gross
4
5
attendance
amount
ticket sales
P
S
5
26420t 1 292,000
5}
ø 0.382
124.02
2407t 1 7220
5.92t 2 131t 1 1000
5 }}
4 }}
2
2
6.02t 2 125t 1 1000
26420t 1 292,000
5.92t 2 2 131t 1 1000
5.92(7)2 2 131(7) 1 1000
2407(7) 1 7220
2(27.8)
c. The paint can is the most efficient, then the coffee can,
}}
+ }}
2
6.02(7) 2 125(7) 1 1000
247,060
and the soup can is the least efficient. The lower the
efficiency ratio, the more that can be put into the can,
while using less material to make the can.
373.08
5}
+}
ø 50.21
419.98
4371
53. S 5 4:r 2 1 2:r*
The average amount a person paid per ticket in 1999 was
$50.21.
k1 + H2 + H + V 2
k1 + HV 2
hg
k1H3V2
}}
}
50. a. } 5 }
5
5
k2
hr
k2 + H2
k2H2
k1HV 2
4
V 5 }3:r3 1 :r 2*
4: r 2 1 2:r*
S
1
r1 }3r 1 * 2
3r 1 }3r 1 * F
2
k2
Mixed Review for TAKS
1
54. A;
V2 5 }
kH
The shorter runner has the advantage. The larger the
height the smaller the fraction representing velocity.
4
51. a. Vsphere 5 }:r 3
3
6(2r 1 * )
r (4r 1 3* )
3y[4 2 ( y 1 2)] 1 5y( y 2 3)
5 3y[2y 1 2] 1 5y 2 2 15y
5 23y 2 1 6y 1 5y 2 2 15y
5 2y 2 2 9y
Vcylinder 5 :r2h
55. H;
4
3
}:r 3 5 :r 2h
0.16x 5 6800
x 5 42,500
3
} + }2 5 h
About 42,500 students attend the University of Texas.
4
3
}r 5 h
Graphing Calculator Activity 8.4 (p. 581)
b. Ssphere 5 4:r
2
1 2
4
Scylinder 5 2:r 2 1 2:rh 5 2:r 2 1 2:r }3r
8
14
5 2:r 2 1 }3 :r 2 5 }
:r 2
3
Ssphere
6
4:r 2
c. } 5 }
5 }7
14 2
Scylinder
}:r
3
Because 6 < 7, the spherical tank uses less material.
2:r 2 1 2:rh
S
52. a. } 5 }
V
:r 2h
2:r(r 1 h)
2(r 1 h)
5}
5}
: +r+r+h
rh
2(r 1 h)
2(3.4 1 10.2)
S
b. Soup can: } 5 } 5 }}
rh
(3.4)(10.2)
V
2(13.6)
5}
ø 0.784
34.68
436
3[2(2r 1 * )]
2
5}
5}
5}
4
4
k1HV 2 5 k2
:r
:r
2: r(2r 1 * )
} 5 }} 5 }
4
4
V
} : r 3 1 : r 2*
: r 2 }3 r 1 *
3
2(2r 1 * )
b. } 5 1
k2
4
3
2(8.4 1 19.4)
5}
ø 0.341
162.96
6.02t 2 125t 1 1000
For 1999, t 5 7:
2(r 1 h)
S
5}
5 }}
Paint can: }
rh
(8.4)(19.4)
V
5 }}
+ }}
2
2407t 1 7220
26420(7) 1 292,000
2(7.8 1 15.9)
2(23.7)
A
4
2(r 1 h)
S
5}
5 }}
Coffee can: }
V
rh
(7.8)(15.9)
Algebra 2
Worked-Out Solution Key
x(x 2 5)
x
x2 2 5x
1. }
5 }}
5}
x22
(x 2 5)(x 2 2)
x2 2 7x 1 10
X
1
2
3
4
5
6
7
X=1
Y1
-1
ERROR
3
2
ERROR
1.5
1.4
Y2
-1
ERROR
3
2
1.6667
1.5
1.4
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
49.
continued
Chapter 8,
continued
3x(x 1 2)
3x
3x 2 1 6x
2. }
5 }}
5}
x24
(x 2 4)(x 1 2)
x 2 2 2x 2 8
X
-2
-1
0
1
2
3
4
X=-2
Y1
ERROR
.6
0
-1
-3
-9
ERROR
Y2
1
.6
0
-1
-3
-9
ERROR
4x 2 2 8x x 1 3
x22
4x 2 2 8x
5. } 4 } 5 } + }
x13
5x 1 15 x 2 2
5x 1 15
4x(x 2 2)(x 1 3)
4x
5 }}
5}
5
5(x 1 3)(x 2 2)
X
-3
-2
-1
0
1
2
3
X=-3
Y1
ERROR
-1.6
-.8
0
.8
ERROR
2.4
Y2
-2.4
-1.6
-.8
0
.8
1.6
2.4
(x 1 1)(x 1 4)
x11
x 2 1 5x 1 4
3. }
5 }}
5}
x23
(x 1 4)(x 2 3)
x 2 1 x 2 12
X
-4
-3
-2
-1
0
1
2
X=-4
Y1
ERROR
.33333
.2
0
-.3333
-1
-3
Y2
.42857
.33333
.2
0
-.3333
-1
-3
x 2 2 3x 2 10 x2 1 2x 2 3
6. }
+}
x2 1 3x 1 3
x2 1 x 2 2
(x 2 5)(x 1 2)
(x 1 3)(x 2 1)
(x 1 2)(x 2 1)
5 }}
+ }}
2
x 1 3x 1 3
(x 2 5)(x 1 2)(x 1 3)(x 2 1)
(x 2 5)(x 1 3)
(x 1 3x 1 3)(x 1 2)(x 2 1)
x 1 3x 1 3
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
5 }}}
5 }}
2
2
(x 1 3)(x 2 1)
x21
x13 x21
4. }
+}
5 }}
5}
x13
5x 2(x 1 3)
5x 2
5x 2
X
-3
-2
-1
0
1
2
3
X=-3
Y1
ERROR
-2.4
-.4
0
0
.8
3.6
X
-2
-1
0
1
2
3
4
X=-2
Y1
ERROR
-12
-5
ERROR
-1.154
-.5714
-.2258
Y2
-7
-12
-5
-2.286
-1.154
-.5714
-.2258
Y2
-7.2
-2.4
-.4
0
0
.8
3.6
Lesson 8.5
8.5 Guided Practice (pp. 582–585)
5
725
2
1
7
1. } 2 } 5 } 5 } 5 }
12x
12x
12x
6x
12x
3
1
211
1
2
2. }2 1 }2 5 }
5 }2 5 }2
3x
3x 2
3x
x
3x
3x
x
4x 2 x
4x
3. } 2 } 5 } 5 }
x22
x22
x22
x22
2(x 2 1 1)
2x 2 1 2
2
2x 2
4. }
1}
5}
5}
52
2
2
2
x 11
x 11
x2 1 1
x 11
5. 5x 3
10x 2 2 15x 5 5x(2x 2 3)
LCM 5 5x 3(2x 2 3)
Algebra 2
Worked-Out Solution Key
437