Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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A-32 APPENDIX V Deriving the Equation of a Conic
A-32
College Algebra G&M—
Appendix V
Deriving the Equation of a Conic
The Equation of an Ellipse
In Section 8.2, the equation was devel-
oped using the distance formula and the definition of an ellipse. To find the standard
form of the equation, we treat this result as a radical equation, isolating one of the rad-
icals and squaring both sides.
isolate one radical
square both sides
We continue by simplifying the equation, isolating the remaining radical, and
squaring again.
simplify
isolate radical; divide by 4
square both sides
expand and distribute
a
2
on left
add 2
a
2
cx
and rewrite equation
factor
divide by
Since , we know and . For convenience, let
(it also follows that and , since ). Substituting b
2
for we
obtain the standard form of the equation of an ellipse (major axis horizontal, since we
stipulated ): . Note once again the x-intercepts are ( , 0), while
the y-intercepts are (0, ). For the foci, to allow for a major axis that
may be vertical.
The Equation of a Hyperbola
In Section 8.3, the equation was devel-
oped using the distance formula and the definition of a hyperbola. To find the standard
form of this equation, we apply the same procedures as before.
isolate one radical
square both sides
simplify
isolate radical; divide by 4
square both sides
expand and distribute
a
2
on the right
add 2
a
2
cx
and rewrite equation
factor
divide by a
2
1c
2
⫺ a
2
2
x
2
a
2
⫺
y
2
c
2
⫺ a
2
⫽ 1
x
2
1c
2
⫺ a
2
2⫺ a
2
y
2
⫽ a
2
1c
2
⫺ a
2
2
c
2
x
2
⫺ a
2
x
2
⫺ a
2
y
2
⫽ a
2
c
2
⫺ a
4
c
2
x
2
⫺ 2a
2
cx ⫹ a
4
⫽ a
2
x
2
⫺ 2a
2
cx ⫹ a
2
c
2
⫹ a
2
y
2
c
2
x
2
⫺ 2a
2
cx ⫹ a
4
⫽ a
2
31x ⫺ c2
2
⫹ y
2
4
cx ⫺ a
2
⫽ a21x ⫺ c2
2
⫹ y
2
4 cx ⫽ 4a
2
⫹ 4a21x ⫺ c2
2
⫹ y
2
x
2
⫹ 2cx ⫹ c
2
⫹ y
2
⫽ 4a
2
⫹ 4a21x ⫺ c2
2
⫹ y
2
⫹ x
2
⫺ 2cx ⫹ c
2
⫹ y
2
1x ⫹ c2
2
⫹ y
2
⫽ 4a
2
⫹ 4a21x ⫺ c2
2
⫹ y
2
⫹ 1x ⫺ c2
2
⫹ y
2
21x ⫹ c2
2
⫹ y
2
⫽ 2a ⫹ 21x ⫺ c2
2
⫹ y
2
21x ⫹ c2
2
⫹ y
2
⫺ 21x ⫺ c2
2
⫹ y
2
⫽ 2a
c
2
⫽
冟
a
2
⫺ b
2
冟
⫾b
⫾a
x
2
a
2
⫹
y
2
b
2
⫽ 1a 7 b
a
2
⫺ c
2
c 7 0a 7 ba
2
7 b
2
b
2
⫽ a
2
⫺ c
2
a
2
⫺ c
2
7 0a
2
7 c
2
a 7 c
a
2
1a
2
⫺ c
2
2
x
2
a
2
⫹
y
2
a
2
⫺ c
2
⫽ 1
x
2
1a
2
⫺ c
2
2⫹ a
2
y
2
⫽ a
2
1a
2
⫺ c
2
2
a
2
x
2
⫺ c
2
x
2
⫹ a
2
y
2
⫽ a
4
⫺ a
2
c
2
a
2
x
2
⫺ 2a
2
cx ⫹ a
2
c
2
⫹ a
2
y
2
⫽ a
4
⫺ 2a
2
cx ⫹ c
2
x
2
a
2
31x ⫺ c2
2
⫹ y
2
4⫽ a
4
⫺ 2a
2
cx ⫹ c
2
x
2
a21x ⫺ c2
2
⫹ y
2
⫽ a
2
⫺ cx
4 cx ⫽ 4a
2
⫺ 4a21x ⫺ c2
2
⫹ y
2
x
2
⫹ 2cx ⫹ c
2
⫹ y
2
⫽ 4a
2
⫺ 4a21x ⫺ c2
2
⫹ y
2
⫹ x
2
⫺ 2cx ⫹ c
2
⫹ y
2
1x ⫹ c2
2
⫹ y
2
⫽ 4a
2
⫺ 4a21x ⫺ c2
2
⫹ y
2
⫹ 1x ⫺ c2
2
⫹ y
2
21x ⫹ c2
2
⫹ y
2
⫽ 2a ⫺ 21x ⫺ c2
2
⫹ y
2
21x ⫹ c2
2
⫹ y
2
⫹ 21x ⫺ c2
2
⫹ y
2
⫽ 2a
expand
binomials
expand
binomials
cob19545_app_032-033.qxd 12/21/10 8:04 PM Page 32
From the definition of a hyperbola we have , showing and
. For convenience, let and substitute to obtain the
standard form of the equation of a hyperbola (transverse axis horizontal): .
Note the x-intercepts are (0, ) and there are no y-intercepts. For the foci,
.
The Asymptotes of a Central Hyperbola
From our work in Section 8.3, a central hyperbola with a horizontal axis will have
asymptotes at . To understand why, recall that for asymptotic behavior we
investigate what happens to the relation for large values of x, meaning as . Starting
with , we have
clear denominators
isolate term with
y
factor out
b
2
x
2
from right side
divide by
a
2
take square roots of both sides
As , and we find that for large values of x, .y ⬇ ⫾
b
a
x
冟
x
冟
Sq,
a
2
x
2
S 0
y ⫽⫾
b
a
x
B
1 ⫺
a
2
x
2
y
2
⫽
b
2
a
2
x
2
a1 ⫺
a
2
x
2
b
a
2
y
2
⫽ b
2
x
2
a1 ⫺
a
2
x
2
b
a
2
y
2
⫽ b
2
x
2
⫺ a
2
b
2
b
2
x
2
⫺ a
2
y
2
⫽ a
2
b
2
x
2
a
2
⫺
y
2
b
2
⫽ 1
冟
x
冟
Sq
y ⫽⫾
b
a
x
c
2
⫽ a
2
⫹ b
2
⫾ a
x
2
a
2
⫺
y
2
b
2
⫽ 1
b
2
⫽ c
2
⫺ a
2
c
2
⫺ a
2
7 0
c
2
7 a
2
0 6 a 6 c
APPENDIX V Deriving the Equation of a Conic A-33
College Algebra G&M—
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A-34 APPENDIX VI Proof Positive—A Selection of Proofs from College Algebra
A-34
College Algebra G&M—
Appendix VI
Proof Positive—A Selection of Proofs
from College Algebra
Proofs from Chapter 4
The Remainder Theorem
If a polynomial p(x) is divided by ( ) using synthetic division, the remainder is
equal to p(c).
Proof of the Remainder Theorem
From our previous work, any number c used in synthetic division will occur as the fac-
tor ( ) when written as .
Here, q(x) represents the quotient polynomial and r is a constant. Evaluating p(c) gives
✓
The Factor Theorem
Given a polynomial p(x),
(1) if , then is a factor of p(x), and
(2) if is a factor of p(x), then .
Proof of the Factor Theorem
1. Consider a polynomial p written in the form . From the
remainder theorem we know , and substituting p(c) for r in the equation
shown gives
and is a factor of p(x), if ,
✓
2. The steps from part 1 can be reversed, since any factor ( ) of p(x), can be written
in the form . Evaluating at produces a result of zero:
✓
Complex Conjugates Theorem
Given p(x) is a polynomial with real number coefficients, complex solutions must occur
in conjugate pairs. If is a solution, then must also be a solution.
To prove this for polynomials of degree , we let and
be complex numbers, and let , and represent their conjugates,
and observe the following properties:
z
2
⫽ c ⫺ diz
1
⫽ a ⫺ bi
z
2
⫽ c ⫹ diz
1
⫽ a ⫹ bin 7 2
a ⫺ bia ⫹ bi, b ⫽ 0
⫽ 0
p1c2⫽ 1c ⫺ c2q1x2
x ⫽ cp1x2⫽ 1x ⫺ c2q1x2
x ⫺ c
p1x2⫽ 1x ⫺ c2q1x2
p1c2⫽ 0x ⫺ c
p1x2⫽ 1x ⫺ c2q1x2⫹ p1c2
p1c2⫽ r
p1x2⫽ 1x ⫺ c2q1x2⫹ r
p1c2⫽ 0x ⫺ c
x ⫺ cp1c2⫽ 0
⫽ r
⫽ 0
#
q1c2⫹ r
p1c2⫽ 1c ⫺ c2q1c2⫹ r
p1x2⫽ 1x ⫺ c2q1x2⫹ r
1quotient21divisor2⫹ remainder: p1x2⫽ 1x ⫺ c2q1x2⫹ rx ⫺ c
x ⫺ c
cob19545_app_034-036.qxd 12/21/10 8:05 PM Page 34
1. The conjugate of a sum is equal to the sum of the conjugates.
✓
2. The conjugate of a product is equal to the product of the conjugates.
✓
Since polynomials involve only sums and products, and the complex conjugate of
any real number is the number itself, we have the following:
Proof of the Complex Conjugates Theorem
Given polynomial , where
a
1
, a
0
are real numbers and is a zero of p, we must show that is
also a zero.
evaluate
p
(
x
) at
z
given
conjugate both sides
property 1
property 2
conjugate of a real number is the number
✓ result
An immediate and useful result of this theorem is that any polynomial of odd
degree must have at least one real root.
Linear Factorization Theorem
If p(x) is a complex polynomial of degree , then p has exactly n linear factors
and can be written in the form , where
and c
1
, c
2
, , c
n
are complex numbers. Some factors may have multiplicities greater
than 1 (c
1
, c
2
, , c
n
are not necessarily distinct).
Proof of the Linear Factorization Theorem
Given is a complex polynomial, the Funda-
mental Theorem of Algebra establishes that p(x) has a least one complex zero, call
it c
1
. The factor theorem stipulates must be a factor of P, giving
where q
1
(x) is a complex polynomial of degree .
Since q
1
(x) is a complex polynomial in its own right, it too must also have a com-
plex zero, call it c
2
. Then ( ) must be a factor of q
1
(x), giving
where q
2
(x) is a complex polynomial of degree .
Repeating this rationale n times will cause p(x) to be rewritten in the form
where q
n
(x) has a degree of , a nonzero constant typically called a
n
.
The result is , and the proof is complete.p1x2⫽ a
n
1x ⫺ c
1
21x ⫺ c
2
2
#
p
#
1x ⫺ c
n
2
n ⫺ n ⫽ 0
p1x2⫽ 1x ⫺ c
1
21x ⫺ c
2
2
#
p
#
1x ⫺ c
n
2q
n
1x2
n ⫺ 2
p1x2⫽ 1x ⫺ c
1
21x ⫺ c
2
2q
2
1x2
x ⫺ c
2
n ⫺ 1
p1x2⫽ 1x ⫺ c
1
2q
1
1x2
1x ⫺ c
1
2
p1x2⫽ a
n
x
n
⫹ a
n⫺1
x
n⫺1
⫹
p
⫹ a
1
x ⫹ a
0
p
p
a
n
⫽ 0p1x2⫽ a
n
1x ⫺ c
1
21x ⫺ c
2
2
#
p
#
1x ⫺ c
n
2
n ⱖ 1
p1z
2⫽ 0
a
n
1z
n
2⫹ a
n⫺1
1z
n⫺1
2⫹
p
⫹ a
1
1z
1
2⫹ a
0
⫽ 0
a
n
1z
n
2⫹ a
n⫺1
1z
n⫺1
2⫹
p
⫹ a
1
1z
1
2⫹ a
0
⫽ 0
a
n
z
n
⫹ a
n⫺1
z
n⫺1
⫹
p
⫹ a
1
z
1
⫹ a
0
⫽ 0
a
n
z
n
⫹ a
n⫺1
z
n⫺1
⫹
p
⫹ a
1
z
1
⫹ a
0
⫽ 0
p 1z2⫽ 0 a
n
z
n
⫹ a
n⫺1
z
n⫺1
⫹
p
⫹ a
1
z
1
⫹ a
0
⫽ 0
a
n
z
n
⫹ a
n⫺1
z
n⫺1
⫹
p
⫹ a
1
z
1
⫹ a
0
⫽ p1z2
z
⫽ a ⫺ biz ⫽ a ⫹ bi
a
n
, a
n⫺1
,
p
,p1x2⫽ a
n
x
n
⫹ a
n⫺1
x
n⫺1
⫹
p
⫹ a
1
x
1
⫹ a
0
1ac ⫺ bd2⫺ 1ad ⫹ bc2i→ conjugate of product → 1ac ⫺ bd2⫹ 1ad ⫹ bc2i
ac ⫺ adi ⫺ cbi ⫹ bdi
2
ac ⫹ adi ⫹ bci ⫹ bdi
2
1a ⫺ bi2
#
1c ⫺ di21a ⫹ bi2
#
1c ⫹ di2
product of conjugates: z
1
#
z
2
product: z
1
#
z
2
1a ⫹ c2⫺ 1b ⫹ d2i→ conjugate of sum → 1a ⫹ c2⫹ 1b ⫹ d2i
1a ⫺ bi2⫹ 1c ⫺ di21a ⫹ bi2⫹ 1c ⫹ di2
sum of conjugates: z
1
⫹ z
2
sum: z
1
⫹ z
2
APPENDIX VI Proof Positive—A Selection of Proofs from College Algebra A-35
College Algebra G&M—
cob19545_app_034-036.qxd 12/21/10 9:38 PM Page 35
Proofs from Chapter 5
The Product Property of Logarithms
Given M, N, and are positive real numbers,
.
Proof of the Product Property
For and , we have and in exponential form.
It follows that
substitute
b
P
for
M
and
b
Q
for
N
properties of exponents
log property 3
substitute log
b
M
for
P
and log
b
N
for
Q
The Quotient Property of Logarithms
Given M, N, and b 1 are positive real numbers,
.
Proof of the Quotient Property
For and , we have and in exponential form.
It follows that
substitute
b
P
for
M
and
b
Q
for
N
properties of exponents
log property 3
substitute log
b
M
for
P
and log
b
N
for
Q
The Power Property of Logarithms
Given M, N, and are positive real numbers and any real number x,
.
Proof of the Power Property
For , we have in exponential form. It follows that
substitute
b
P
for
M
properties of exponents
log property 3
substitute log
b
M
for
P
rewrite factors ⫽ x log
b
M
⫽ 1log
b
M2x
⫽ Px
⫽ log
b
1b
Px
2
log
b
1M2
x
⫽ log
b
1b
P
2
x
b
P
⫽ MP ⫽ log
b
M
log
b
M
x
⫽ x log
b
M
b ⫽ 1
⫽ log
b
M ⫺ log
b
N
⫽ P ⫺ Q
⫽ log
b
1b
P⫺Q
2
log
b
a
M
N
b⫽ log
b
a
b
P
b
Q
b
b
Q
⫽ Nb
P
⫽ MQ ⫽ log
b
NP ⫽ log
b
M
log
b
a
M
N
b⫽ log
b
M ⫺ log
b
N
⫽
⫽ log
b
M ⫹ log
b
N
⫽ P ⫹ Q
⫽ log
b
1b
P⫹Q
2
log
b
1MN2⫽ log
b
1b
P
b
Q
2
b
Q
⫽ Nb
P
⫽ MQ ⫽ log
b
NP ⫽ log
b
M
log
b
1MN2⫽ log
b
M ⫹ log
b
N
b ⫽ 1
A-36 APPENDIX VI Proof Positive—A Selection of Proofs from College Algebra
College Algebra G&M—
cob19545_app_034-036.qxd 12/21/10 8:05 PM Page 36
SA-1
CHAPTER R
Exercises R.1, pp. 8–10
1. constant 3. coefficient 5. Answers will vary
7. two; 3 and 9. two; 2 and 11. three; 1, and
13. one; 15. 17. 19. 21.
23. 25. 27. 29. Let w represent the
width in meters. Then 2w represents twice the width and
represents three meters less than twice the width. 31. Let b represent the
speed of the bus. Then represents 15 mph more than the speed of
the bus. 33. 35. 37.
39. 41. 14 43. 19 45. 0 47. 16 49.
51. 51 53. 2 55. 144 57. 59. 24
61. 63.
gives an output of 0. 3 gives an output of 0.
65.
2 gives an output of 0.
67. a. b. c.
d. 69. a. 3.2 b. 71.
73. 75. 77. 79. 81.
83. 85. 87. 89. 10 ohms
91. a. b. 93. a. b. 107 ft
95. 44¢ 97. $81
99. a. positive odd integer
Exercises R.2, pp. 21–24
1. power 3. 20x;0 5. a. cannot be simplified, unlike terms
b. can be simplified, like bases 7. 9. 11.
13. 15. 17. 19. 21.
23. 25. a. b. 27. 29.
31. 33. 35. 37. 39. 41. 43.
45. 47. 49. 51. 53. 55.
57. 59. 2 61. 63. 65. 67.
69. 0.000 000 006 5 71. 26,571 hrs; 1107 days 73. polynomial, none
of these, degree 3 75. nonpolynomial because exponents are not whole
numbers, NA, NA 77. polynomial, binomial, degree 3
6.77 ⫻ 10
9
⫺4
13
9
7
10
⫺2b
7
27a
9
c
3
⫺12
5x
4
a
12
b
4
c
8
1
a
3
5
3h
7
3p
2
⫺4q
2
25p
2
q
2
4
25m
4
n
6
4r
8
8x
6
27y
9
4p
8
q
6
⫺8
⫺1
8
2h
3
27
8
⫺3ab3w
3
1728 units
3
V ⫽ 27x
6
9
4
x
3
y
2
9
16
x
6
y
2
49c
14
d
4
p
2
4q
2
32.768h
3
k
6
216p
3
q
6
a
14
b
7
⫺12p
5
q
4
14n
7
C ⫽ 25t ⫹ 43.50;t ⫽ c ⫹ 29;
L ⫽ 2W ⫹ 3t ⫽ 275 mpht ⫽
1
2
j
7a
2
⫺ 13a ⫺ 5
29
8
n ⫹
38
5
2a ⫹ 3b ⫹ 2c
6x
2
⫺ 3x⫺2a
2
⫹ 2a
17
12
x⫺2a⫺
2
15
p ⫹ 6
⫺5x ⫹ 13
5
6
x ⫹ 7 ⫺ 7 ⫽ x
a ⫹ 1⫺4.22⫹ 13.6 ⫽ a ⫹ 9.4n ⫹ 1⫺227 ⫹ 1⫺52⫽ 2
⫺1
⫺
41
5
⫺36T ⫽ 12.50g ⫹ 50
M ⫽ 2.5NL ⫽ 2W ⫹ 20h ⫽ b ⫹ 150
b ⫹ 15
2w ⫺ 3
31n ⫹ 52⫺ 7
2
3
n ⫺ 5n
2
⫹ 2n
2n ⫺ 131n ⫺ 52
2
n ⫹ 4n ⫺ 7⫺1
⫺5⫺2,
1
4
⫺5
79. 81.
83. 85. 87.
89. 91.
93. 95. 97.
99. 101. 103.
105. 107. 109.
111. 113.
115. 117.
119. 121.
123. 125. 127.
129. 131.
133. 135.
137. a. 340 mg, 292.5 mg b. Less, amount is decreasing. c. after 5 hr
139. 141. 143. $15 145. 6
Exercises R.3, pp. 35–39
1. identity; unknown; contradiction; unknown 3. simple; compound
5. Answers will vary. 7. 9. 11.
13. 15. 17. 19. 21.
23. 25. 27. 29.
31. contradiction; { } 33. conditional;
35. identity; 37.
39.
41.
;
43.
;
45.
;
47. { } 49. 51.
53.
55.
57.
59.
61. ;
63. no solution
65. ;
67.
69.
)(
0 0.5⫺0.5 1⫺1
x 僆 1⫺
1
3
, ⫺
1
4
2;
[
[
0 1⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
x 僆 3⫺5, 04;
⫺2 ⫺1 0 321 4⫺3⫺4
x 僆 1⫺q, q2
)[
1 2 3 54 60⫺3 ⫺1⫺2
x 僆 3⫺2, 52
)
)
⫺2 ⫺10 213⫺3⫺4
x 僆 1⫺q,⫺22 ´ 11, q2;
54, 66; 52, 4, 5, 6, 7, 86
5 6; 5⫺3, ⫺2, ⫺1, 0, 1, 2, 3, 4, 5, 6, 76
526; 5⫺3, ⫺2, ⫺1, 0, 1, 2, 3, 4, 6, 86
5p
|
p ⑀ ⺢65x
|
x ⑀ ⺢6
x 僆 1⫺q,
⫺32
5
2
)
⫺4⫺3 ⫺20⫺1⫺5⫺10⫺9 ⫺6⫺7⫺8
5x
|
x 6
⫺32
5
6;
n 僆 31, q2
2130
[
5n
|
n ⱖ 16;
a 僆 32, q2
3452⫺3 ⫺210⫺1
[
5a
|
a ⱖ 26;
5x0⫺2 ⱕ x ⱕ 16; 3⫺2, 14
5x0x ⱖ ⫺26; 3⫺2, q25x0x 僆 ⺢6
n ⫽⫺
11
10
p ⫽
12
5
n ⫽
20
21
v ⫽⫺0.5a ⫽⫺3.6
p ⫽⫺56x ⫽ 12x ⫽ 12m ⫽⫺
27
4
b ⫽⫺15
b ⫽
6
5
v ⫽⫺11x ⫽ 3
5x
⫺3
⫹ 3x
⫺2
⫹ 2x
⫺1
⫹ 4F ⫽ kPQd
⫺2
k
3
⫹ 3k
2
⫺ 28k ⫺ 60xy ⫹ 2x ⫺ 3y ⫺ 6
16 ⫺ 81x
⫹ x16p
2
⫺ 24pq ⫹ 9q
2
16g
2
⫹ 24g ⫹ 9x
2
⫹ 8x ⫹ 16x ⫺ 16; x
2
⫺ 6
36 ⫺ 25k
2
6 ⫺ 5k;49x
2
⫺ 1007x ⫹ 10;
16m
2
⫺ 94m ⫹ 3;2x
4
⫺ x
2
⫺ 15
12c
2
⫹ 23cd ⫹ 5d
2
6x
2
⫹ 11xy ⫺ 10y
2
m
2
⫺
9
16
x
2
⫹
3
4
x ⫹
1
8
p
2
⫹ 1.1p ⫺ 9
9 ⫺ m
2
21v
2
⫺ 47v ⫹ 20b
3
⫺ b
2
⫺ 34b ⫺ 56
x
3
⫺ 273r
2
⫺ 11r ⫹ 10⫺3x
3
⫹ 3x
2
⫹ 18x
q
6
⫹ q
5
⫺ q
4
⫹ 2q
3
⫺ q
2
⫺ 2q
1
4
x
2
⫺ 8x ⫹ 6
7.85b
2
⫺ 0.6b ⫺ 1.93p
3
⫺ 3p
2
⫺ 12
⫺2
3
x
2
⫹ 12;
⫺2
3
c
3
⫹ 2c
2
⫺ 3c ⫹ 6; 1⫺w
3
⫺ 3w
2
⫹ 7w ⫹ 8.2; ⫺1
Student Answer Appendix
x Output
14
6
0
0
1
2
3 ⫺4
⫺6
⫺6
⫺4
⫺1
⫺2
⫺3
x Output
0
1
2
30
⫺3
⫺6
⫺9
⫺12⫺1
⫺15⫺2
⫺18⫺3
x Output
8
9
04
1
20
313
⫺1
⫺1
⫺2
⫺5⫺3
cob19545_saa_001-009.qxd 11/23/10 3:29 PM Page SA-1
71.
73.
75.
77.
79. a. ✓ b. ✓ c. 20 vartices
81. 83.
85. 87.
89. 91. , at least 81%
93. , at least $2000
95.
97.
99. more than 6 hr
101. a. 216.5 ft
2
b. 7 sheets will be needed 103. about 337.4 in
3
105. 107. Answers may vary. 109. 111. 113.
115.
Exercises R.4, pp. 49–53
1. product 3. binomial; conjugate 5. Answers will vary.
7. a. b. c.
9. a. b. c.
11. a. b. c.
13. a. b. prime c.
15. a. b. c.
17. a. b. c.
d. e. 19. a.
b. c. d.
21. a. b.
c. d.
23. a. b.
c.
25. a. b.
c. d.
27. 29. 31.
33. 35. 37.
39. 41. 43.
45. 47. prime 49.
51. a. H b. E c. C d. F e. B f. A g. I h. D i. G
53. 55. not quadratic
57. 59.
61. not quadratic (degree 3) 63.
65. or 67. 69. or
71. or 73. or 75.
77. or or 79. or
81. or 83. or
85. or 87. or
89. 91.
93. , , 95. , ,
97. , , , 99. , , ,
101.
103.
105. a. 3 in. b. 5 in. c. V ⫽ 2412921272⫽ 18,792 in
3
V ⫽ x1x ⫹ 521x ⫹ 32
6 cm
3
; 18.8 cm
3
V ⫽
1
3
h1R ⫹ r21R ⫺ r2;
21,991 cm
2
2r 1r ⫹ h2,7000 cm
2
;
b ⫽ 5b ⫽ 4b ⫽⫺1b ⫽⫺2x ⫽ 5x ⫽ 2x ⫽⫺2x ⫽⫺5
x ⫽ 5x ⫽ 2x ⫽⫺5p ⫽ 3p ⫽⫺3p ⫽⫺7
x ⫽⫺3, x ⫽ 0, x ⫽
2
3
x ⫽⫺2, x ⫽ 0, x ⫽ 11
w ⫽ 3w ⫽⫺
1
2
x ⫽⫺3x ⫽ 5
t ⫽ 2t ⫽⫺13r ⫽⫺3r ⫽ 8
c ⫽ 15c ⫽⫺3m ⫽ 3m ⫽⫺3m ⫽⫺5
g ⫽⫺9a ⫽⫺3a ⫽ 3h ⫽
⫺1
2
h ⫽ 0
p ⫽ 2p ⫽ 0m ⫽ 4x ⫽⫺3x ⫽ 5
a ⫽ 1; b ⫽⫺1; c ⫽⫺5
a ⫽ 2; b ⫽ 0; c ⫽ 7a ⫽
1
4
; b ⫽⫺6; c ⫽ 0
1a ⫽ 02a ⫽⫺1; b ⫽ 2; c ⫽⫺15
1x ⫺ 521x ⫹ 321x ⫺ 3212x ⫺ 5214x
2
⫹ 10x ⫹ 252
1a ⫹ 521a ⫺ 1224m1m ⫹ 521m ⫺ 22⫺3x12x ⫺ 721x ⫺ 32
13k ⫺ 42
2
12h ⫹ 321h ⫹ 221r ⫺ 321r ⫺ 62
⫺113m ⫹ 8213m ⫺ 8221x ⫺ 221x ⫺ 1021a ⫹ 521a ⫹ 22
7x12x ⫹ 1212x ⫺ 121n ⫹ 121n
2
⫺ n ⫹ 12
1n ⫺ 121n
2
⫹ n ⫹ 121n ⫹ 121n ⫺ 12
1x ⫺ 221x
2
⫹ 2x ⫹ 421x ⫹ 121x
2
⫺ x ⫹ 12
1x
2
⫹ 921x
2
⫹ 421x ⫹ 321x ⫺ 321x ⫹ 121x ⫺ 12
⫺2t1t ⫺ 321t
2
⫹ 3t ⫹ 921g ⫺ 0.321g
2
⫹ 0.3g ⫹ 0.092
1m ⫹
1
2
21m
2
⫺
1
2
m ⫹
1
4
212p ⫺ 3214p
2
⫹ 6p ⫹ 92
13n ⫺ 72
2
12m ⫺ 52
2
1b ⫹ 52
2
1a ⫺ 32
2
1b ⫹ 2521b ⫺ 252111h ⫹ 122111h ⫺ 122
215x ⫹ 6215x ⫺ 6213x ⫹ 7213x ⫺ 7212s ⫹ 5212s ⫺ 52
15u ⫹ 3212u ⫺ 5214q ⫺ 521q ⫹ 3213p ⫹ 221p ⫺ 52
1n ⫺ 421n ⫺ 52⫺11p ⫺ 721p ⫹ 22
1k
2
⫺ 721k
3
⫺ 521h ⫺ 1221h
4
⫺ 3213q ⫹ 2213q
2
⫹ 52
1n ⫹ 7214m ⫺ 1121b
2
⫹ 3213b ⫹ 221a ⫹ 2212a ⫹ 32
⫺3a
2
1a
2
⫹ 2a ⫺ 327b13b
2
⫺ 2b ⫹ 82⫺171x
2
⫺ 32
7
666x ⫽
1
2
S ⫽ 4.5 h ⫹ 20, K ⫽ 6 h ⫹ 11,
45 6
9
5
C ⫹ 32 6 85, 7.2° 6 C 6 29.4°
0 6 20 w 6 150, 0 6 w 6 7.5 m
1125 ⫹ 850 ⫹ 625 ⫹ 400 ⫹ x
5
ⱖ 1000
82 ⫹ 76 ⫹ 65 ⫹ 71 ⫹ x
5
ⱖ 7574 ⫽
10
3
r ⫹ 34; 12 in.
41 ⫽
2
3
S ⫹ 1; 60 yr2c ⫹ 13 ⫽ 1467; 727 cal
2f ⫹ 5 ⫽ 73; 34 fansW 6 177.34 lb
6 ⫹ 8 ⫺ 12 ⫽ 28 ⫹ 6 ⫺ 12 ⫽ 2
)[
04812⫺20 ⫺8 ⫺4⫺16⫺12
x 僆 3⫺16, 82;
⫺1.4
[
0.8
012⫺2 ⫺1
[
x 僆 3⫺1.4, 0.84;
0⫺1123⫺5⫺6 ⫺4 ⫺3 ⫺2
[
)
x 僆 3⫺4, 12;
⫺2 ⫺101 324⫺3⫺4
x 僆 1⫺q, q2;
107. ,
109. 11 in. by 13 in.
111. a. b.
113. 115.
117.
119.
Exercises R.5, pp. 61–64
1. 1; 3. common denominator 5. F; numerator should be
7. a. b. 9. a. simplified b.
11. a. b. 13. a. b. c. d.
15. a. b. c. d. 17.
19. 1 21. 23. 25. 27. 29.
31. 33. 35. 37.
39. 41. 43. 45.
47. 49. 51. 53.
55. 57. 59.
61. 63. a.
b. 65. 67. 69.
71. 73. a. b. 75.
77. 79. 81. is extraneous 83.
85.
87. a. $300 million; $2550 million 89. Price rises rapidly for first four
b. It would require many resources. days, then begins a gradual
c. No decrease. Yes, on the 35th day
of trading.
91. 93.
95. b.
12x ⫹ 3y2
5
P ⬇ 80%t ⫽ 8 weeks
a ⫽⫺1, a ⫽⫺8
n ⫽ 7x ⫽ 3; x ⫽ 7y ⫽ 12a ⫽
3
2
x ⫽ 1
1 ⫹
2
x
2
1 ⫺
2
x
2
;
x
2
⫹ 2
x
2
⫺ 2
1 ⫹
3
m
1 ⫺
3
m
;
m ⫹ 3
m ⫺ 3
⫺2
y ⫹ 31
x
313x ⫺ 42
p ⫺ 1
4a
a ⫹ 20
1
x
2
⫹
2
x
3
;
x ⫹ 2
x
3
1
p
2
⫺
5
p
;
1 ⫺ 5p
p
2
y
2
⫹ 26y ⫺ 1
15y ⫹ 121y ⫹ 321y ⫺ 22
m
2
⫺ 6m ⫹ 21
1m ⫹ 32
2
1m ⫺ 32
1
y ⫹ 1
2a ⫺ 5
1a ⫹ 421a ⫺ 52
⫺y ⫹ 11
1y ⫹ 621y ⫺ 52
⫺5m ⫹ 37
m ⫺ 7
⫺3m ⫺ 16
1m ⫹ 421m ⫺ 42
2
p ⫹ 6
14y ⫺ x
8x
2
y
4
3 ⫹ 20x
8x
2
2n ⫹ 1
n
31a
2
⫹ 3a ⫹ 92
2
n ⫹
1
5
n ⫹
2
3
x ⫹ 0.3
x ⫺ 0.2
y ⫹ 3
3y1y ⫹ 42
m
m ⫺ 4
y
x
81a ⫺ 72
a ⫺ 5
3
2
⫺15
4
1p ⫺ 42
2
p
2
1a ⫺ 221a ⫹ 12
1a ⫹ 321a ⫹ 22
n ⫺ 2x ⫹ 2
3x ⫹ 5
2x ⫹ 3
2n ⫹ 3
n
⫺1
m
⫺11y ⫹ 32
x ⫹ 3
9
⫺3ab
9
⫺1⫺1
a ⫺ 4
a ⫺ 7
x ⫹ 3
2x1x ⫺ 22
⫺
1
3
⫺1⫺1
1q ⫹ 521q ⫺ 521q ⫹ 2321q ⫺ 232
1p ⫹ 121p
2
⫺ p ⫹ 121p ⫺ 121p
2
⫹ p ⫹ 12
1x ⫹ 321x ⫺ 321x
2
⫹ 922x116x ⫺ 27216x ⫹ 52
1
18
112b
5
⫺ 3b
3
⫹ 8b
2
⫺ 182
1
8
14x
4
⫹ x
3
⫺ 6x
2
⫹ 322
⫽ 327 in. ⬇ 7.94 in.
L ⫽ 12211 ⫹ 0.75211 ⫺ 0.752
L ⫽ L
0
B
a1 ⫹
v
c
b a1 ⫺
v
c
b
SA-2 Student Answer Appendix
Day Price
0 10
1 16.67
2 32.76
3 47.40
4 53.51
5 52.86
6 49.25
7 44.91
8 40.75
9 37.03
10 33.81
P
40 300
60 675
80 1800
90 4050
93 5979
95 8550
98 22050
100 ERROR
450P
100 ⫺ P
cob19545_saa_001-009.qxd 11/23/10 3:29 PM Page SA-2
Exercises R.6, pp. 76–80
1. even 3. 5. Answers will vary. 7. a. b.
9. a. b. c. d. 11. a. 4 b. c.
d. 13. a. 2 b. not a real number c. d. e.
f. 15. a. b. c. not real number d.
17. a. 4 b. c. d. 19. a.
b. not a real number c. d. 21. a. b.
23. a. b. c. d.
e. f. 25. a. b. c.
d. 27. a. b. c. d.
29. a. b. c. d. e.
31. a. b. c. d.
33. a. b.
c. 35. a. 98 b.
c. d.
37. a. b.
c.
39. Verified 41. Verified 43. a. b. c.
d. e. 45. a. b.
47. a. 0.05
b. 49. a.
b. is extraneous 51. a. b. c.
d. 53. a. b.
c. d. is extraneous 55. a.
b. 57. a. b.
59. , 61. 8.33 ft 63. a. b. about 25.3 m
65. a. 365.02 days b. 688.69 days c. 87.91 days 67. a. 36 mph
b. 46.5 mph 69. 71. a. 36 million mi
b. 67 million mi c. 93 million mi d. 142 million mi e. 484 million mi
f. 887 million mi 73. a. b.
75. a. b. Answers will vary. 77.
79.
Practice Test, pp. 83–84
1. a. False; parentheses first b. False; undefined c. True
d. False; 2. a. 11 b. c. not a real number d. 20
3. a. b. c. 0.5 d. 4. a. b. 0.9 c. 4 d.
5. 6. a. 0 b. undefined 7. a.
b. 8. a. b. 9. a.
b. 10. a. Let r represent Earth’s radius. Then
represents Jupiter’s radius. b. Let e represent this year’s earnings. Then
million represents last year’s earnings. 11. a.
b. c. 12. a. b.
c. 13. a. b. c. d.
14. a. b. c. d.
15. a. b.
16. a. b. 17. a.
b. c. d. e. f.
31m 72
51m 421m 32
x 5
3x 1
x 5
3x 2
x 3
2 n
2 n
17x
4
4x
2
5x7a
4
5a
3
8a
2
3a 18
4a
2
12ab 9b
2
9x
4
25y
2
6
a
12
b
4
c
8
6.4 10
2
0.0644ab
25
4
p
2
q
2
m
6
8n
3
4a
12
b
12
5b
3
1x 521x 321x 32
v12v 32
2
13x 4213x 42x
2
6x7b 8
9v
2
3v 74e 1.2
11r 1192n 3a
n
2
b
2
x
3
12x 92⬇ 7.29132;
1
3
, 1
3; 2, 6, 5⬇ 4439.28
7
28
3
4.6
7
6
9
8
52x 6
x 僆 31, 22 ´ 12, q2
322
2
1313x 391x
1n 11921n 11921x 1521x 152
12234 ⬇ 219.82 m
2
8210 mx 125x 27
x 30, x 34x 32, x 22x 81
x 32x 6; x 0x 2, x 18
x 7; x 2 is extraneousx 25x 16
m 64x 5m 3x 8, x 1
x
14
3
7.60
7 712
16 213
3
;
130
215 313 312;
61x
612
x 2
12 4111
; 1.27
52
3
a
2
a
2
3
2p
2
2p
326b
10b
2215x
9x
2
23
2
1215
2114 36115 6142
110 165 217 118219
39 1213
n
2
5
115
1216x22x 522 27x 323
2 13x 315x1
3
2x
517p1612m1413912
b
3
4
1
1
6
6
2
6
6
5
6
1
12
bx
2
1
4
x2x
2
y
3
181
3
3
z
3
315
4x
3n2m
2
3u
2
v1
3
v
x
4
1y
3
4b1b
15a
2
9
2
12
3 17
4pq
3
12p
3
2
mn2
3
n
2
10pq
2
1
3
q3m12
1
2y
1
4
32n
10
p
2
256
81x
4
1
9
1728
9p
4
4q
2
125
8
64
125
7
冟
v
5
冟
6
3
冟
n
3
冟
5
冟
h 2
冟
k 33x3x
2
v
2
6z
4
6x
冟
x 3
冟
9m
2
冟
x 3
冟
7
冟
p
冟
冟
10
冟
10
冟
9
冟
9116
1
4
2
3
Student Answer Appendix SA-3
18. a. b. c. d. e.
f. g. h. 19.
a. 10 decreases of 0.50 or $5.00 b. Maximum revenue is $1250.
20. 58 cm 21. a. b. c. d.
e. (contradiction) f.
22. the number is 40
23. a. b.
c. 24. a. b.
c. (2 is extraneous) 25. a.
b. c. (7 is extraneous)x 1x 4, x 5
x 3, x 3n 13
h
5
3
, h 2x
1
2
g 3, g 1, g 1, g 3
r 0, r 1, r 4x 2, x 2, x 7
3
1
4
1n 122 16,
g 105 6
x
1
6
m 1n 4b 6
0.5x
2
10x 1200;2116 122
110x
5x
x
2
5
11110
1
2
12
2
64
125
2
3v
冟
x 11
冟
CHAPTER 1
Exercises 1.1, pp. 98–102
1. first, second 3. radius, center 5. Answers will vary.
7.
9.
11.
13. 15. 17.
19. 21. 23.
x
y
108642
10
8
6
4
2
1
2
3
4
5
2
3
4
5
1
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
R: y 僆 ⺢R: y 僆 ⺢R: 0 y 5
D: x 僆 ⺢D: x 1D: 5 x 5
x
y
54321
5
4
3
2
1
3
6
9
12
15
6
9
12
15
3
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
R: y 1
D: x 僆 ⺢
D 54, 1, 2, 36; R 50, 5, 4, 2, 36
D 51, 3, 5, 7, 96; R 52, 4, 6, 8, 106
R 52.75, 3.00, 3.25, 3.50, 3.756
D 51, 2, 3, 4, 56
1
2
3
4
5
Year in
college
2.75
3.00
3.25
3.50
3.75
GPA
xy
5
3
01
3
6
8
13
3
3
1
3
6
xy
0
0 2,
1 3,
3 5,
6 8,
7 9, 9
8
5
3
2
2
xy
8
3
0
23
38
415
1
2
3
xy
3
4
05
2
34
43
121
3
4
xy
10 3,
5 2,
4
2 1,
1.25 0.5,
1 0
0.5
1
13
, 13
2
3
xy
0
0 1
4
7
2
1
3
5
1
12
29
cob19545_saa_001-009.qxd 11/24/10 6:00 PM Page SA-3
25.
27.
29. 31. 33. 35. 37.
39. 41. 10 43. right triangle 45. not a right triangle
47. right triangle
49. 51.
53. 55.
57. 59.
61. 63.
65. 67. (2, 3),
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
r 2, x 僆 30, 44, y 僆 31, 5 41x 52
2
1y 42
2
9
x
y
1512963
15
12
9
6
3
3
6
9
12
15
6
9
12
15
3
x
y
20161284
20
16
12
8
4
4
8
12
16
20
8
12
16
20
4
1x 32
2
1y 42
2
411x 72
2
1y 12
2
100
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
1x 42
2
1y 52
2
121x 12
2
1y 22
2
9
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
1x 72
2
1y 42
2
71x 42
2
1y 32
2
4
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
1x 52
2
y
2
3x
2
y
2
9
2234
11, 0210, 121
1
20
,
1
24
210.7, 0.3213, 12
10
10
10
10
10, 02, 10, 42
10
10
10
10
10, 22, 15, 02
69.
71.
73. , (5, 6),
75.
77.
79.
81.
83.
85. a. 87. b.
89. f. 91. j.
93. a. (3, 183), (5, 241), (7, 299), (9, 357), (11, 415); yes b. $473
c. 2014 d.
95. a. b. no
97. Red: ;
Blue: ;
Area blue 12 units
2
1x 22
2
y
2
16
1x 22
2
1y 22
2
4
1x 52
2
1y 122
2
625
(11, 415)
(3, 183)
678021345
200
300
91011
x
y
400
6x y x
2
91x 12
2
1y 22
2
49
y x
2
1y 32
2
36y x
2
6x
x
y
1512963
15
12
9
6
3
3
6
9
12
15
6
9
12
15
3
1x 32
2
1y 52
2
32, 13, 52, r 412
x
y
1512963
15
12
9
6
3
3
6
9
12
15
6
9
12
15
3
1x 72
2
y
2
37, 17, 02, r 137
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
1x 22
2
1y 52
2
11, 12, 52, r 111
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
x
2
1y 32
2
14, 10, 32, r 114
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
1x 52
2
1y 22
2
25, 15, 22, r 5
x
y
1512963
15
12
9
6
3
3
6
9
12
15
6
9
12
15
3
r 1571x 52
2
1y 62
2
57
x
y
1512963
15
12
9
6
3
3
6
9
12
15
6
9
12
15
3
14, 02, r 9, x 僆 313, 54, y 僆 39, 94
y 僆 32 213, 2 2134
11, 22, r 213, x 僆 31 213, 1 2134,
SA-4 Student Answer Appendix
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
cob19545_saa_001-009.qxd 11/24/10 6:00 PM Page SA-4
99.
101. , Answers will vary.
103. Answers will vary.
105. a. center: (degenerate case) b. center: (1, 4);
c. ; degenerate case 107.
109. is a solution, is extraneous
Exercises 1.2, pp. 112–116
1. 0, 0 3. negative, downward 5.
7. 9.
11. 13.
✓
✓
15. 17. 19.
21. 23. 25.
27. 29. 31.
33. ; 35. ;
x
y
84
8
4
4
8
4
8
(10, 3)
(4, 5)
x
y
84
8
4
4
8
4
8
(4, 6)
(3, 5)
17, 12 and 11, 92 12, 42 and 11, 32
m
4
3
m 1
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(3, 0)
(0, 4)
(6, 4)
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(0, 0)
(2, 3)
(4, 6)
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(5, 0)
(0, 2)
(5, 4)
x
y
84
8
4
40
80
40
80
(0, 25)
(1, 75)
(
,
0
)
1
2
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(2, 1)
(4, 2)
(0, 0)
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(0, 4)
(3, 6)
(6, 0)
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(4, R)
(2, 0)
(0,
R)
(
T
,
0)
(0, 3)
(2, 8)
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(5, 0)
(0, 1)
(5, 2)
19
4
19
4
19
4
3
4
4
19
4
3
2
1
1
2
2 4
0.5 0.5
0.5
9
2
4
(0, 6)
(1, 3)
(2, 0)
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
0.5
3
2
132 4
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(0, 4)
(2, 1)
(2, 7)
(4, 10)
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(0, 2)
(3, 4)
(6, 6)
(3, 0)
yes no
m
1
m
2
m
1
#
m
2
1
n 2n 1
x
7
4
r
2
1
r 516, 22; r 0
y 4.8; y 3.6
x
y
1512963
15
12
9
6
3
3
6
9
12
15
6
9
12
15
3
37. ; 39.
;
and
41. a. , cost increased $125,000 per 1000 sq ft b. $375,000
43. a. , distance increases 22.5 mph b. about 186 mi
45. a. , a person weighs 23 lb more for each additional 6 in. in
height b.
47. In inches: (0, ) and (576, ): . The sewer line is 1 in.
deeper for each 48 in. in length.
49. 51.
53. L
1
: ; L
2
: ; point of intersection (2, 4)
55. a. For any two points chosen , indicating
there has been no increase or decrease in the number
of supreme court justices. b. For any two points
chosen , which indicates that over the last 5
decades, one nonwhite or nonfemale justice has been
added to the court every 10 yr.
57. parallel 59. neither 61. parallel 63. right triangle
65. not a right triangle 67. right triangle 69. a. 78.2 yr b. 2015
71. a. $3500 b. 5 yr
73. a. 273 in. b. 20 months
75. Yes they will meet, the two roads are not parallel: .
77. a. $7360 b. 2015 79. a. 21% b. 2016
81. 83. a. 142 b. c. 9 d.
85. a. b. 4 87. a.
b. c.
Exercises 1.3, pp. 127–131
1. first 3. range 5. Answers will vary. 7. function
9. Not a function. The Shaq is paired with two heights.
11. Not a function; 4 is paired with 2 and . 13. function
15. function 17. Not a function; is paired with 3 and .
19. function 21. function 23. Not a function; 0 is paired with 4 and .
25. function 27. Not a function; 4 is paired with and 1.
29. function
31. 33.
function function
35. 37.
39.
41.
43.
45.
47. 49. 5x
|
2 x 16; 32, 145x
|
x 26; 32, q2
x 僆 12, 52
)
)
234 657143012
x 僆 1q, 12 ´ 11, q2
)
)
123 54604 3 12
m 僆 1q, 54
[
123 54604 3 12
p 僆 1q, 32
)
123403 2 1
T 僆 1250, 4502250 6 T 6 450;w 僆 345, q2w 45;
y
2
1
1
2
3
4
5
6
7
3
x
321
7
6
5
4
3
2
1
x
y
54321
5
4
3
2
1
1
2
3
4
5
2
3
4
5
1
1
4
42
5
1x 521x
2
5x 2521x 2421x 12
1x 321x 221x 221015
27
2
83a 6
38
12
30
9.5
h 3t 300
v 1250t 8500
m
1
10
m 0
y 4x 2
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(2, 6)
(2, 0)
(2, 4)
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(3, 4)
(3, 0)
(3, 5)
m
1
48
186
⬇3.8
m
23
6
m 22.5
m 125
x
y
84
8
4
4
8
4
8
(4, 2)
(3, 6)
x
y
84
8
4
4
8
4
8
(3, 7)
(1, 8)
110, 102 and 111, 2217, 22215, 232
m
4
7
m
15
4
Student Answer Appendix SA-5
xy
6
4
02
30
3
6
xy
1
04
27
410
2
Time
t
Nonwhite, nonmale
Justices
30 40 605010 20
1
2
3
4
5
6
7
8
9
10
No, distance between
centers is less
than sum of radii.
cob19545_saa_001-009.qxd 11/24/10 6:01 PM Page SA-5