Document 1804502

Chapter 8
Prerequisite Skills (p. 548)
3. Answers will vary. The equation should be of the form
xy 5 a, where a is the approximate value of each product
in Exercise 2.
1. The asymptote of the graph is y 5 0.
2. Two variables x and y show direct variation provided
y 5 ax where a is a nonzero constant.
3. An extraneous solution of a transformed equation is not
an actual solution of the original equation.
4. y 5 ax
y 5 4x
8 5 a(2)
5 4(22)
45a
5 28
4. Answers will vary.
8.1 Guided Practice (pp. 552–554)
1. y 5 3x
x and y show direct variation because the equation is of
the form y 5 ax.
2. xy 5 0.75
y 5 4x
5.
0.75
y 5 ax
y5}
x
y 5 24x
4 5 a(21)
24 5 a
5 24(22)
x and y show inverse variation because the equation is of
58
the form y 5 }x .
y 5 24x
6.
a
3. y 5 x 2 5
1
y 5 }6 x
y 5 ax
x and y show neither direct variation nor inverse variation
a
because the equation is not of the form y 5 ax or y 5 }x .
1
2 5 a(12)
5 }6 (22)
4.
2
2
12
5 2}6
1
6
5 2}3
}5a
1
y 5 }6 x
8. 2x 3 2 4x 2 1 2x 5 2x(x 2 2 2x 1 1)
6.
21 5 }8
5 2}2
12 5 a
56
28 5 a
5 24
8
5 }2
65a
53
2
6
y 5 }x
5 2(3x 2 2)(x 3 2 4)
5 10x 2 x 2 6
11. (22x 2 1 6) 2 (x 2 2 x) 5 22x 2 1 6 2 x 2 1 x
5 23x 2 1 x 1 6
12. (x 1 2)(x 2 9) 5 (x 1 2)[x 2 2 2(x)(9) 1 92]
2
5 (x 1 2)(x 2 2 18x 1 81)
5 (x 1 2)x 2 1 (x 1 2)(218x) 1
(x 1 2)(81)
5 x 3 1 2x 2 2 18x 2 2 36x 1 81x 1 162
5 x 3 2 16x 2 1 45x 1 162
Lesson 8.1
Investigating Algebra Activity 8.1 (p. 550)
1. No, because the ratios of apparent height to the distance
are not approximately equal.
2. Answers will vary. The products are approximately equal.
6
}
2
2
y 5 }x
a
5 2[x 3(3x 2 2) 2 4(3x 2 2)]
10. (3x 2 6) 1 (7x 2 x) 5 3x 2 1 7x 2 2 x 2 6
6
a
12 5 }
1
9. 6x 4 2 4x 3 2 24x 1 16 5 2(3x 4 2 2x 3 2 12x 1 8)
2
y 5 2}x
y 5 }x
5 2x(x 2 1)(x 2 1)
5 2x(x 2 1)2
8
a
12
8
y 5 2}x
5}
2
12
7. x 2 11x 2 26 5 (x 2 13)(x 1 2)
y 5 ax
3 5 }4
y5}
x
2
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
5.
y5}
x
a
1
}5a
12
y 5 ax
a
7.
n 5 }s
a
3000 5 }5
15,000 5 a
15,000
A model is n 5 }
.
s
26,000
26,000
8. c 5 } 5 } ø 329
79
A
The number of chips per wafer for a chip with an area of
79 square millimeters is about 329.
9. z 5 axy
7
z 5 }2 xy
7
7 5 a(1)(2)
5 }2(22)(5)
7 5 2a
5 235
7
}5a
2
7
z 5 }2 xy
Algebra 2
Worked-Out Solution Key
415
Chapter 8,
continued
10.
z 5 22xy
z 5 axy
24 5 a(4)(23)
5 22(22)(5)
24 5 212a
5 20
2
7. y 5 }
x
x and y show inverse variation because the equation is of
a
the form y 5 }x .
22 5 a
8. x 1 y 5 6
z 5 22xy
z 5 axy
y 5 2x 1 6
3
z 5 2}2 xy
18 5 a(22)(6)
3
5 2}2 (22)(5)
18 5 212a
5 15
x and y show neither direct nor inverse variation because
a
the equation is not of the form y 5 ax or y 5 }x .
9. 8y 5 x
x
3
y 5 }8
2}2 5 a
x and y show direct variation because the equation is of
the form y 5 ax.
3
z 5 2}2 xy
12.
z 5 axy
7
10. xy 5 12
z 5 }3 xy
12
7
56 5 a(26)(24)
5 }3 (22)(5)
56 5 24a
5 2}
3
y5}
x
x and y show inverse variation because the equation is of
70
a
the form y 5 }x .
7
3
}5a
11. C; xy 5 5
7
z 5 }3 xy
aw
13. x 5 }
y
5
y 5 }x
aqr
14. p 5 }
s
y 5 }x
20
24 5 }5
Skill Practice
20
y 5 2}
x
said to vary jointly with x and y.
2. Find the product x + y for each data pair (x, y). If the
products are constant or approximately constant, then
the set of data pairs shows inverse variation.
24
y 5 }x
24
a
a
the form y 5 }x .
21
y 5 }x
21
4
1 2
x and y show direct variation because the equation is of
the form y 5 ax.
5
y 5 }x
2
y 5 }x
y 5 }x
a
1
2}6 5 }
212
1
2}6 (212) 5 a
25a
2
y 5 }x
Algebra 2
Worked-Out Solution Key
2}4(24) 5 a
55a
a
18.
a
5
57
21
y5}
x
6. 4x 5 y
x and y show direct variation because the equation is of
the form y 5 ax.
5
2}4 5 }
24
21 5 a
y 5 8x
y 5 }x
5}
3
3
28 }4 5 a
a
x
14
5}
3
a
17.
y5}
x
a
because the equation is not of the form y 5 ax or y 5 }.
14
y5}
x
14
}
x and y show neither direct variation nor inverse variation
a
y 5 }x
y5}
x
28 5 }
3
4. y 5 x 1 4
53
14 5 a
5 28
a
16.
95a
a
24
y 5 2}
x
x and y show inverse variation because the equation is of
5 }3
2 5 }7
5 2}
3
224 5 a
1
y5}
5x
416
15.
y 5 2}
x
85}
23
1
3. xy 5 }
5
9
9 5 }1
9
y 5 }x
a
14.
y 5 }x
a
5 2}
3
220 5 a
1. If z varies directly with the product of x and y, then z is
9
a
13. y 5 }
x
y 5 2}
x
a
8.1 Exercises (pp. 555–557)
y
5. } 5 8
x
20
a
12.
2
5 }3
5
y 5 }x
5
5 }3
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
11.
Chapter 8,
continued
35
a
19.
y 5 }x
26.
y 5 2}
3x
1 2
1
15 5 a 2}4 (23)
35
a
27 5 }
5
5 2}
3(3)
}
3
z 5 20xy
z 5 axy
3
15 5 }4 a
35
5 2}
9
5
27 }3 5 a
1 2
5 2400
20 5 a
35
5a
2}
3
z 5 20xy
27.
35
2}
3
1
z 5 axy
z5}
xy
14
35
y5}
5 2}
3x
x
1
23 5 a(16)(27)
(24)(5)
5}
14
23 5 242a
5 2}
14
20. x + y: 1.5(40) 5 60, 2.5(24) 5 60, 4(15) 5 60,
7.5(8) 5 60, 10(6) 5 60
y 40
x 1.5
80
3
48
5
24
2.5
15
4
16
15
6
10
3
5
1
28.
21. x + y: 12(132) 5 1584, 18(198) 5 3564,
23(253) 5 5819, 29(319) 5 9251, 34(374) 5 12,716
253
23
319
29
374
34
y
x and y show direct variation because the ratios }x
}: } 5 11, } 5 11, } 5 11, } 5 11, } 5 11
9
7
6
11
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
x and y show neither direct variation nor inverse variation
y
because neither the products x + y nor the ratios }x
are equal.
23. x + y: 4(21) 5 84, 6(14) 5 84, 8(10.5) 5 84,
6 5 218a
5}
3
20
1
z 5 axy
z 5 25xy
75 5 a(5)(23)
5 25(24)(5)
75 5 215a
5 100
25 5 a
z 5 25xy
30. A; z 5 axy
236 5 a(23)(24)
y 21
7
21
10.5
14
}: } 5 5.25, } 5 } ø 2.3, } 5 } ø 1.3,
3
16
x 4
6
8
236 5 12a
10
8.4
25
21
7
12
x and y show inverse variation because the products x + y
are equal.
z 5 axy
z 5 22xy
24 5 a(2)(26)
5 22(24)(5)
24 5 212a
5 40
22 5 a
z 5 22xy
25.
5 2}3 (24)(5)
8.4(10) 5 84, 12(7) 5 84
} 5 } ø 1.2, } ø 0.6
24.
1
6 5 a(9)(22)
z 5 2}3 xy
29.
10
6.2
11
5
}: } 5 4, } 5 2.2, } ø 1.6, } ø 1.3, } ø 0.5
z 5 2}3 xy
1
22. x + y: 4(16) 5 64, 5(11) 5 55, 6.2(10) 5 62, 7(9) 5 63,
11(6) 5 66
1
z 5 axy
2}3 5 a
are equal.
y 16
x 4
5 2}
7
z5}
xy
14
x and y show inverse variation because the products x + y
are equal.
198
18
10
}5a
} 5 } ø 1.1, } 5 } 5 0.6
y 132
x 12
20
1
14
}: } 5 } ø 26.7, } 5 } 5 9.6, } 5 3.75,
8
7.5
z 5 axy
1
z 5 }4 xy
1
12 5 a(8)(6)
5 }4 (24)(5)
12 5 48a
5 215
1
}5a
4
5 20(24)(5)
23 5 a
ay
31. x 5 }
z
32. y 5 axz 2
axz
33. w 5 }
y
34. The variables x and y should be in the numerator and the
variable w should be in the denominator because z varies
directly with x and y and inversely with w. The correct
axy 3
equation is z 5 }
}.
Ïw
2
2
35. Sample answer: f(x) 5 2x, g(x) 5 }, h(x) 5 2x 1 };
x
x
2
h(2) 5 2(2) 1 }2 5 4 1 1 5 5
36. x varies directly with z.
a
b
x 5 }y , y 5 }z
a
a
x5}
5 }b z 5 cz
b
}
z
1
z 5 }4 xy
Algebra 2
Worked-Out Solution Key
417
continued
Problem Solving
37.
(3.53 3 1022)(2.25 3 1022) 5 G(1.19002 3 1055)
7.9425 3 1044 5 G (1.19002 3 1055)
103.68
n5}
s
a
n 5 }s
7.9425 3 1044
1.19002 3 10
103.68
a
54 5 }
1.92
103.68 5 a
5}
3.87
}}
55 5 G
ø 26.8
7.9425 10
} 5G
1}
1.19002 21 1055 2
44
103.68
A model is n 5 }
. You can store 26 photos on your
s
camera when the average photo size is 3.87 megapixels.
38. Sample answer: 7.4(1200) 5 8880, 8.9(1000) 5 8900,
6.7 3 10211 ø G
c. As the masses of the two objects increase and
the distance between them is held constant, the
gravitational force increases. As the masses of the two
objects are held constant and the distance between
them increases, the gravitational force decreases.
12.1(750) 5 9075, 17.9(500) 5 8950
9000
I + R 5 9000 or R 5 }
I
9000
9000
R5}
5}
ø 265
I
34
When I 5 34 milliamps, the resistance is about 265 ohms.
172
a
39.
P5}
A
P5}
A
a
0.43 5 }
400
172
5}
60
172 5 a
ø 2.87
aWD 2
42. a. P 5 }
L
a(2W )D 2
aWD 2
}5}5P
L
2L
P stays the same when the width and length of the
beam are doubled.
aWD 2
b. P 5 }
L
a(2W)(2D)2
8aWD 2
} 5 } 5 8P
L
L
172
P5}
A
P is multiplied by 8 when the width and depth of the
beam are doubled.
The pressure if you wear the boots is about 2.87 pounds
per square inch.
aWD 2
c. P 5 }
L
a(2W )(2D)2
8aWD 2
4aWD 2
} 5 } 5 } 5 4P
L
2L
2L
}
40. a.
aÏ T
f5}
Ld
}
aÏ 670
262 5 }
62(0.1025)
P is quadrupled when all three dimensions
are doubled.
}
1665.01 5 aÏ 670
64.3 ø a
d. Sample answer: If the safe load of a beam is increased
}
64.3Ï T
by a factor of 4, you can double the length, width,
and depth, or you can quadruple the width, or you can
double the depth.
f5}
Ld
}
}
64.3ÏT
64.3Ï 1629
b. f 5 } 5 } ø 26.3
Ld
201.6(0.49)
The frequency of the note is about 26.3 hertz.
43. C;
Gm1m2
41. a. F 5 }
d2
a2 1 b2 5 c2
a 2 1 92 5 182
Gm1m2
b. F 5 }
d2
a 2 1 81 5 324
G(5.98 3 10
)(1.99 3 10 )
(1.50 3 10 )
G(5.98 3 1.99)(1024 3 1030)
3.53 3 1022 5 }}}
(1.50)2(1011)2
G(11.9002 3 1054)
22
24
30
3.53 3 1022 5 }}}
11 2
3.53 3 10 5 }}
22
2.25 3 10
G(1.19002 3 1055)
3.53 3 1022 5 }}
22
2.25 3 10
418
Mixed Review for TAKS
Algebra 2
Worked-Out Solution Key
a 2 5 243
a ø 15.588
1
A 5 }2bh
1
ø }2 (15.588)(9)
5 70.1
The area of nMNP is about 70.1 square centimeters.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Chapter 8,
Chapter 8,
continued
2x 1 1
5. y 5 }
4x 2 2
44. F;
The domain is all real numbers
On their own, 8 hexagons and 6 squares have
8(6) 1 6(4) 5 72 edges.
y
In the solid, each edge is shared by exactly two polygons.
1
1
except }2, and the range is all
1
real numbers except }2 .
1
So, the number of edges is }2 (72) 5 36.
x
1
F1V5E12
14 1 V 5 36 1 2
23x 1 2
6. f(x) 5 }
2x 2 1
V 5 24
The domain is all real numbers
except 21, and the range is all
real numbers except 3.
y
The solid has 24 vertices.
Lesson 8.2
8.2 Guided Practice (pp. 559–561)
4
1. f (x) 5 2}
x
The domain is all real numbers
except 0, and the range is all
real numbers except 0.
y
1
x
1
x
22
7. If the cost of the 3-D printer is $21,000, then the function
300m 1 21,000
300m 1 21,000
is c 5 }}
. The graph of c 5 }}
m
m
lies closer to the axes than the graph of
21
300m 1 24,000
. Both graphs have the same
c 5 }}
m
asymptotes, domain, and range.
8
2. y 5 } 2 5
x
The domain is all real numbers
except 0, and the range is all
real numbers except 25.
y
2
x
22
8.2 Exercises (pp. 561–563)
Skill Practice
7
1. The function y 5 } 1 3 has a range of all real
x14
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
numbers except 3 and a domain of all real numbers
except 24.
23x 1 5
2. The function f (x) 5 }
is not a rational function
2x 1 1
because the expression 2x 1 1 is not a polynomial.
1
3. y 5 } 1 2
x23
The domain is all real numbers
except 3, and the range is all
real numbers except 2.
y
3
3. y 5 }
x
The graph lies farther from the
1
x
21
1
10
4. y 5 }
x
x
21
1
axes than the graph of y 5 }x .
y
The graph lies farther from the
1
axes than the graph of y 5 }x .
y
x21
4. y 5 }
x13
The domain is all real numbers
except 23, and the range is all
real numbers except 1.
y
Both graphs lie in the first and
third quadrants and have the same
asymptotes, domain, and range.
Both graphs lie in the first and
third quadrants and have the same
asymptotes, domain, and range.
2
x
22
2
21
x
25
5. y 5 }
x
The graph lies farther from the
1
axes than the graph of y 5 }x and it
y
1
21
x
lies in Quadrants II and IV instead
of Quadrants I and III. Both
graphs have the same asymptotes,
domain, and range.
Algebra 2
Worked-Out Solution Key
419