Sat math Essentials

18.
135
The length of an arc is equal to the circumfer-
ence of the circle multiplied by the measure of
the angle that intercepts the arc divided by
360. The arc measures 15
π
units, the circum-
ference of a circle is 2
π
multiplied by the
radius, and the radius of the circle is 20 units. If
x
represents the measure of angle 
AOB
, then:
15
π
= 
36
x
0
2
π
(20)
15 = 
36
x
0
(40)
15 = 
9
x
x
= 135
The measure of angle 
AOB
is 135 degrees.
Section 3 Answers
1.
d.
2
5
= 0.40.
3
7
≈
0.43. Comparing the hun-
dredths digits, 3 > 0, therefore, 0.43 > 0.40
and 
3
7
> 
2
5
.
2.
b.
Solve 3
x
– 
y
= 2 for 
y
: –
y
= –3
x
+ 2,
y
= 3
x
–
2. Substitute 3
x
– 2 for 
y
in the second equa-
tion and solve for 
x
:
2(3
x
– 2) – 3
x
= 8
6
x
– 4 – 3
x
= 8
3
x
– 4 = 8
3
x
= 12
x
= 4
Substitute the value of
x
into the first equation
to find the value of
y
:
3(4) – 
y
= 2
12 – 
y
= 2
y
= 10
x
y
= 
1
4
0
= 
2
5
.
3.
c.
The roots of an equation are the values for
which the equation evaluates to zero. Factor
x
3
+ 7
x
2
– 8
x
:
x
3
+ 7
x
2
– 8
x
= 
x
(
x
2
+ 7
x
– 8) = 
x
(
x
+ 8)(
x
– 1). When 
x
= 0, –8, or 1, the equa-
tion 
f
(
x
) = 
x
3
+ 7
x
2
– 8
x
is equal to zero. The set
of roots is {0, –8, 1}.
4.
b.
First, find the slope of the line. The slope of a
line is equal to the change in 
y
values divided by
the change in 
x
values of two points on the line.
The 
y
value increases by 2 (5 – 3) and the 
x
value decreases by 4 (–2 – 2). Therefore, the
slope of the line is equal to –
2
4
, or –
1
2
. The equa-
tion of the line is 
y
= –
1
2
x
+ 
b
, where 
b
is the
y
-intercept. Use either of the two given points to
solve for 
b
:
3 = –
1
2
(2) + 
b
3 = –1 + 
b
b
= 4
The equation of the line that passes through the
points (2,3) and (–2,5) is 
y
= –
1
2
x
+ 4.
5.
a.
The empty crate weighs 8.16 kg, or 8,160 g. If
Jon can lift 11,000 g and one orange weighs 220
g, then the number of oranges that he can pack
into the crate is equal to
11,00
2
0
2
–
0
8,160
= 
2
2
,8
2
4
0
0
≈
12.9. Jon cannot pack a fraction of an orange.
He can pack 12 whole oranges into the crate.
6.
d.
The volume of a prism is equal to 
lwh
, where 
l
is the length of the prism,
w
is the width of the
prism, and 
h
is the height of the prism:
(2
x
)(6
x
)(5
x
) = 1,620
60
x
3
= 1,620
x
3
= 27
x
= 3
The length of the prism is 2(3) = 6 mm, the
width of the prism is 6(3) = 18 mm, and the
height of the prism is 5(3) = 15 mm.
–
P R A C T I C E T E S T 2
–
2 1 8

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7.
a.
At the start, there are 5 + 3 + 2 = 10 pens in the
box, 3 of which are black. Therefore, the proba-
bility of selecting a black pen is 
1
3
0
. After the black
pen is removed, there are nine pens remaining in
the box, five of which are blue. The probability of
selecting a blue pen second is 
5
9
. To find the proba-
bility that both events will happen, multiply the
probability of the first event by the probability of
the second event: (
1
3
0
)(
5
9
) = 
1
9
5
0
= 
1
6
.
8.
b.
Angle 
CBD
and angle 
PBZ
are alternating
angles—their measures are equal. Angle 
PBZ
=
70 degrees. Angle 
PBZ
+ angle 
ZBK
form angle
PBK
. Line 
PQ
is perpendicular to line 
JK
; there-
fore, angle 
PBK
is a right angle (90 degrees).
Angle 
ZBK
= angle 
PBK
– angle 
PBZ
= 90 – 70
= 20 degrees.
9.
c.
For the first four days of the week, Monica sells
12 pretzels, 12 pretzels, 14 pretzels, and 16 pret-
zels. The median value is the average of the sec-
ond and third values:
12 +
2
14
= 
2
2
6
= 13. If Monica
sells 13 pretzels on Friday, the median will still
be 13. She will have sold 12 pretzels, 12 pretzels,
13 pretzels, 14 pretzels, and 16 pretzels. The
median stays the same.
10.
a.
The denominator of each term in the pattern is
equal to 2 raised to the power given in the
numerator. The numerator decreases by 1 from
one term to the next. Since 10 is the numerator
of the first term, 10 – 9, or 1, will be the numer-
ator of the tenth term. 2
1
= 2, so the tenth term
will be 
1
2
.
11.
a.
No matter whether 
p
is positive or negative, or
whether 
p
is a fraction, whole number, or mixed
number, the absolute value of three times any
number will always be positive and greater than
the absolute value of that number.
12.
d.
Line 
OB
line 
OC
, which means the angles
opposite line 
OB
and 
OC
(angles 
C
and 
B
) are
congruent. Since angle 
B
= 55 degrees, then
angle 
C
= 55 degrees. There are 180 degrees in
a triangle, so the measure of angle 
O
is equal to
180 – (55 + 55) = 180 – 110 = 70 degrees. Angle
O
is a central angle. The measure of its inter-
cepted arc, minor arc 
BC
, is equal to the meas-
ure of angle 
O
, 70 degrees.