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Math Problems
Algebra 1
Scale drawings: word problems
Choose the inequality that represents the following graph. A number line from
−
5
-5
−
5
to
5
5
5
with evenly spaced tick marks in increments of
1
1
1
. A line going to the left starts at
−
3
-3
−
3
and continues left. A closed point is plotted at
−
3
-3
−
3
where the line begins.
\newline
−
5
-5
−
5
−
4
-4
−
4
−
3
-3
−
3
−
2
-2
−
2
−
1
-1
−
1
0
0
0
1
1
1
2
2
2
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A sphere of radius
2
2
2
inches is cut by three planes passing through its center. This partitions the solid into
8
8
8
equal parts, one of which is shown. The volume of each part is
t
π
t \pi
t
π
cubic inches. What is the value of
t
t
t
?
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Sophie drew a scale drawing of a house and its lot. The back patio is
3
3
3
centimeters wide in the drawing. The actual patio is
18
18
18
meters wide. What scale did Sophie use for the drawing?
\newline
1 centimeter : ________ meters
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The diameter of a plate is
12
12
12
inches. Find the circumference (hint:
C
=
3.14
×
d
C= 3.14\times d
C
=
3.14
×
d
). As in the given hint, use
3.14
3.14
3.14
for pi.
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The penguin exhibit at a zoo has a raised circular island that is surrounded by water. The diameter of the island is
20
20
20
meters. One penguin swims half way around the island before hopping out.
\newline
How far did the penguin swim? Give your answer in terms of pi.
\newline
□
\square
□
meters
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A table with a round top is cut in half so that it can be used against a wall. After it is cut, the edge along the wall is
6
6
6
feet.
\newline
What is the area
A
A
A
of the table after it is cut in half ?
\newline
Give your answer in terms of pi.
\newline
A
=
□
A=\square
A
=
□
feet
2
\text {feet }^{2}
feet
2
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A python curls up to touch the tip of its own tail with its nose, forming the shape of a circle. The python is
2.6
π
2.6 \pi
2.6
π
meters long.
\newline
What is the radius
r
r
r
of the circle that the python forms?
\newline
r
=
□
r=\square
r
=
□
m
\mathrm{m}
m
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A solid with surface area
8
8
8
square units is dilated by a scale factor of
k
k
k
to obtain a solid with surface area
A
A
A
square units. Find the value of
k
k
k
which leads to an image with each given surface area.
\newline
PART A
512
512
512
square units
\newline
PART B
(
1
)
(
2
)
\frac{(1)}{(2)}
(
2
)
(
1
)
square unit
\newline
PART C
8
8
8
square units
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Owen made a scale drawing of a city. In real life, a neighborhood park is
170
170
170
yards long. It is
17
17
17
inches long in the drawing. What scale did Owen use for the drawing?
\newline
1
1
1
inch :
_
_
_
\_\_\_
___
yards
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Carmen drew a scale drawing of a house and its lot. In real life, the lawn in the backyard is
72
meters
72\,\text{meters}
72
meters
long. It is
24
millimeters
24\,\text{millimeters}
24
millimeters
long in the drawing. What is the scale of the drawing?
\newline
1
millimeter
:
_
_
_
meters
1\,\text{millimeter} : \_\_\_\,\text{meters}
1
millimeter
:
___
meters
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Gabriel measured a house and made a scale drawing. The garage, which is
12
12
12
meters long in real life, is
4
4
4
millimeters long in the drawing. What scale did Gabriel use for the drawing?
\newline
1
1
1
millimeter :
_
_
\_\_
__
meters
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Camilla drew a scale drawing of a house and its lot. In real life, the back patio is
27
27
27
meters long. It is
3
3
3
millimeters long in the drawing. What scale did Camilla use?
\newline
1
1
1
millimeter :
_
_
_
\_\_\_
___
meters
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Without dividing, determine if
92
92
92
,
842
842
842
is divisible by
4
4
4
and explain how you know.
\newline
92
92
92
,
842
842
842
□
\square
□
divisible by
4
4
4
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Without dividing, determine if
11
11
11
,
653
653
653
is divisible by
3
3
3
and explain how you know.
\newline
11
11
11
,
653
653
653
□
\square
□
divisible by
3
3
3
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In an arithmetic sequence, the first term,
a
1
a_{1}
a
1
, is equal to
5
5
5
, and the third term,
a
3
a_{3}
a
3
, is equal to
9
9
9
. Which number represents the common difference of the arithmetic sequence?
\newline
d
=
2
d=2
d
=
2
\newline
d
=
3
d=3
d
=
3
\newline
d
=
4
d=4
d
=
4
\newline
d
=
5
d=5
d
=
5
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In a geometric sequence, the first term,
a
1
a_{1}
a
1
, is equal to
4
4
4
, and the third term,
a
3
a_{3}
a
3
, is equal to
144
144
144
. Which number represents the common ratio of the geometric sequence?
\newline
r
=
5
r=5
r
=
5
\newline
r
=
6
r=6
r
=
6
\newline
r
=
7
r=7
r
=
7
\newline
r
=
8
r=8
r
=
8
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IQ scores are normally distributed with a mean of
100
100
100
and a standard deviation of
15
15
15
. Out of a randomly selected
1350
1350
1350
people from the population, how many of them would have an IQ between
95
95
95
and
123
123
123
, to the nearest whole number?
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A square with a perimeter of
137
137
137
units is the image of a square that was dilated by a scale factor of
4
4
4
. Find the perimeter of the preimage, the original square, before its dilation. Round your answer to the nearest tenth, if necessary.
\newline
Answer: units
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A square with an area of
134
134
134
units
2
^{2}
2
is dilated by a scale factor of
2
2
2
. Find the area of the square after dilation. Round your answer to the nearest tenth, if necessary.
\newline
Answer: units
2
^{2}
2
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A triangle with a perimeter of
43
43
43
units is the image of a triangle that was dilated by a scale factor of
4
3
\frac{4}{3}
3
4
. Find the perimeter of the preimage, the original triangle, before its dilation. Round your answer to the nearest tenth, if necessary.
\newline
Answer: units
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A rectangle with a perimeter of
126
126
126
units is the image of a rectangle that was dilated by a scale factor of
1
2
\frac{1}{2}
2
1
. Find the perimeter of the preimage, the original rectangle, before its dilation. Round your answer to the nearest tenth, if necessary.
\newline
Answer: units
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A rectangle with an area of
140
140
140
units
2
^{2}
2
is the image of a rectangle that was dilated by a scale factor of
3
4
\frac{3}{4}
4
3
. Find the area of the preimage, the original rectangle, before its dilation. Round your answer to the nearest tenth, if necessary.
\newline
Answer: units
2
^{2}
2
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A rectangle with a perimeter of
43
43
43
units is dilated by a scale factor of
3
4
\frac{3}{4}
4
3
. Find the perimeter of the rectangle after dilation. Round your answer to the nearest tenth, if necessary.
\newline
Answer: units
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The volume of a right cone is
3750
π
3750 \pi
3750
π
units
3
^{3}
3
. If its circumference measures
30
π
30 \pi
30
π
units, find its height.
\newline
Answer: units
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A scientist mixed three chemicals,
R
R
R
,
S
S
S
, and
T
T
T
, in a glass container. The amount of
R
R
R
is
3
3
3
times the amount of
S
S
S
, and the amount of
T
T
T
is
(
1
)
/
(
6
)
(1)/(6)
(
1
)
/
(
6
)
the amount of
S
S
S
. What is the ratio of the amount of
R
R
R
to the amount of
T
T
T
?
\newline
E.
S
S
S
1
1
1
\newline
F.
S
S
S
2
2
2
\newline
G.
S
S
S
3
3
3
\newline
H.
S
S
S
4
4
4
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Topics: Long division
\newline
On dividing
426
426
426
by
4
4
4
, we get the remainder as
\newline
0
0
0
\newline
2
2
2
\newline
4
4
4
\newline
6
6
6
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Polygon
H
H
H
is a scaled copy of Polygon
G
G
G
using a scale factor of
1
4
\frac{1}{4}
4
1
.
\newline
Polygon
H
H
H
's area is what fraction of Polygon
G
′
G^{\prime}
G
′
's area?
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Polygon
F
F
F
has an area of
36
36
36
square units. Aimar drew a scaled version of Polygon
F
F
F
and labeled it Polygon
G
G
G
. Polygon
G
G
G
has an area of
4
4
4
square units.
\newline
What scale factor did Aimar use to go from Polygon
F
F
F
to Polygon
G
G
G
?
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Polygon
Y
Y
Y
is a scaled copy of Polygon
X
X
X
using a scale factor of
1
3
\frac{1}{3}
3
1
.
\newline
Polygon
Y
′
Y^{\prime}
Y
′
's area is what fraction of Polygon
X
X
X
's area?
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Polygon
Q
Q
Q
is a scaled copy of Polygon
P
P
P
using a scale factor of
1
2
\frac{1}{2}
2
1
.
\newline
Polygon
Q
Q
Q
's area is what fraction of Polygon
P
′
s
P^{\prime} \mathrm{s}
P
′
s
area?
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The following data points represent the yearly salaries of high school cheerleading coaches in Dakota County (in thousands of dollars).
\newline
41
,
38
,
36
,
57
,
43
41,38,36,57,43
41
,
38
,
36
,
57
,
43
\newline
Find the mean absolute deviation (MAD) of the data set.
\newline
thousand dollars
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A group of friends went to play minigolf at Adventureland. Anna's score was
26
26
26
putts, Daniela's score was
31
31
31
putts, Luiza's score was
39
39
39
putts, and Vera's score was
32
32
32
putts.
\newline
Find the mean absolute deviation (MAD) of the data set.
\newline
putts
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Cayden has several screws on a scale, and the scale reads
80.955
g
80.955 \mathrm{~g}
80.955
g
. Cayden adds
1
1
1
more screw, and the scale reads
84.81
g
84.81 \mathrm{~g}
84.81
g
.
\newline
What is the mass of the last screw Cayden adds?
\newline
g
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The following are all angle measures (in radians, rounded to the nearest hundredth) whose sine is
0
0
0
.
98
98
98
.
\newline
Which is the principal value of
arcsin
(
0.98
)
?
\arcsin (0.98) ?
arcsin
(
0.98
)?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
11
-11
−
11
.
20
20
20
\newline
(B)
−
4
-4
−
4
.
91
91
91
\newline
(C)
1.37
\mathbf{1 . 3 7}
1.37
\newline
(D)
7
7
7
.
65
65
65
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A pendulum is swinging next to a wall. The distance from the bob of the swinging pendulum to the wall varies in a periodic way that can be modeled by a trigonometric function.
\newline
The function has period
0
0
0
.
8
8
8
seconds, amplitude
6
c
m
6 \mathrm{~cm}
6
cm
, and midline
H
=
15
c
m
H=15 \mathrm{~cm}
H
=
15
cm
. At time
t
=
0.5
t=0.5
t
=
0.5
seconds, the bob is at its midline, moving towards the wall.
\newline
Find the formula of the trigonometric function that models the distance
H
H
H
from the pendulum's bob to the wall after
t
t
t
seconds. Define the function using radians.
\newline
H
(
t
)
=
□
H(t)=\square
H
(
t
)
=
□
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Avery is designing a large rectangular sign. They want the sign to have an area of
2
m
2
2 \mathrm{~m}^{2}
2
m
2
and a width of
4
5
m
\frac{4}{5} \mathrm{~m}
5
4
m
.
\newline
How tall should the sign be?
\newline
m
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S
E
=
σ
n
S E=\frac{\sigma}{\sqrt{n}}
SE
=
n
σ
\newline
The given equation relates the standard error,
S
E
S E
SE
, of a sample mean to the population standard deviation,
σ
\sigma
σ
, and the size of the sample,
n
n
n
. Which of the following equations correctly gives the size of the sample in terms of the standard error and the population standard deviation?
\newline
Choose
1
1
1
answer:
\newline
(A)
n
=
(
σ
S
E
)
2
n=\left(\frac{\sigma}{S E}\right)^{2}
n
=
(
SE
σ
)
2
\newline
(B)
n
=
σ
2
S
E
n=\frac{\sigma^{2}}{S E}
n
=
SE
σ
2
\newline
(C)
n
=
(
σ
S
E
)
n=\sqrt{\left(\frac{\sigma}{S E}\right)}
n
=
(
SE
σ
)
\newline
(D)
n
=
σ
S
E
n=\frac{\sqrt{\sigma}}{S E}
n
=
SE
σ
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With a mass of
112
112
112
.
67
67
67
grams, the Star of India is one of the largest star sapphires in the world. The density of a star sapphire is approximately
4
4
4
.
0
0
0
grams per cubic centimeter. Which of the following best approximates the volume, in cubic centimeters, of the Star of India?
\newline
(Density is mass per unit volume.)
\newline
Choose
1
1
1
answer:
\newline
(A)
26
26
26
\newline
(B)
28
28
28
\newline
(C)
30
30
30
\newline
(D)
32
32
32
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A cake pan is in the shape of a right rectangular prism
20
20
20
centimeters (cm) long by
20
c
m
20 \mathrm{~cm}
20
cm
wide by
5
c
m
5 \mathrm{~cm}
5
cm
high. The pan contains
1
1
1
,
000
000
000
cubic centimeters
(
c
m
3
)
\left(\mathrm{cm}^{3}\right)
(
cm
3
)
of batter.
\newline
Approximately how far is the cake batter from the top of the pan?
\newline
Choose
1
1
1
answer:
\newline
(A)
1
c
m
1 \mathrm{~cm}
1
cm
\newline
(B)
2.5
c
m
2.5 \mathrm{~cm}
2.5
cm
\newline
(C)
5
c
m
5 \mathrm{~cm}
5
cm
\newline
(D)
7
c
m
7 \mathrm{~cm}
7
cm
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The temperature,
T
T
T
, within a rod
x
x
x
centimeters from the rod's left end is modeled by the following equation:
\newline
T
=
−
110
+
0.13
x
(
21
−
x
)
T=-110+0.13 x(21-x)
T
=
−
110
+
0.13
x
(
21
−
x
)
\newline
Both ends of the rod have the same temperature. According to the model, what is the length of the rod in centimeters?
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The surface of a road curves like a parabola to allow water to drain off of it. The given equation shows the surface height,
y
y
y
, in feet above the base at a point
x
x
x
feet from the left edge of the road.
\newline
y
=
−
0.0015
x
(
x
−
40
)
y=-0.0015 x(x-40)
y
=
−
0.0015
x
(
x
−
40
)
\newline
The base has the same width as the road. What is the width of the road in feet?
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A
(
x
)
=
(
2
x
+
8
)
(
2
x
+
10
)
A(x)=(2 x+8)(2 x+10)
A
(
x
)
=
(
2
x
+
8
)
(
2
x
+
10
)
\newline
The function models
A
A
A
, the area, in square inches, of a rectangular framed picture with an
8
8
8
inch by
10
10
10
inch picture and a frame width of
x
x
x
inches. What is the area, in square inches, of the framed picture if the frame is
0
0
0
.
5
5
5
inch wide?
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A circle has a circumference of
4
π
4 \pi
4
π
feet (ft). An arc,
x
x
x
, in this circle has a central angle of
24
0
∘
240^{\circ}
24
0
∘
. What is the length of
x
x
x
?
\newline
Choose
1
1
1
answer:
\newline
(A)
4
π
3
f
t
\frac{4 \pi}{3} \mathrm{ft}
3
4
π
ft
\newline
(B)
8
π
3
f
t
\frac{8 \pi}{3} \mathrm{ft}
3
8
π
ft
\newline
(C)
480
f
t
480 \mathrm{ft}
480
ft
\newline
(D)
960
f
t
960 \mathrm{ft}
960
ft
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A conical glass holds
131
131
131
cubic centimeters of water. It has a height of
5
5
5
centimeters. The radius of the top of the glass is
m
5
π
\sqrt{\frac{m}{5 \pi}}
5
π
m
centimeters, where
m
m
m
is a constant. What is the value of
m
m
m
?
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A cylindrical glass of water is
2
2
2
.
5
5
5
centimeters tall and holds a volume of
20
20
20
cubic centimeters of water.
\newline
The radius of the glass is
p
π
\sqrt{\frac{p}{\pi}}
π
p
centimeters, where
p
p
p
is a constant. What is the value of
p
p
p
?
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The moon has a volume of about
4
3
π
(
1
2
3
)
3
\frac{4}{3} \pi\left(12^{3}\right)^{3}
3
4
π
(
1
2
3
)
3
cubic kilometers. The earth has a volume of about
4
3
π
(
16
⋅
2
0
2
)
3
\frac{4}{3} \pi\left(16 \cdot 20^{2}\right)^{3}
3
4
π
(
16
⋅
2
0
2
)
3
cubic kilometers. The radius of the earth is
u
27
\frac{u}{27}
27
u
times as long as the radius of the moon. What is the value of
u
u
u
?
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Eric made a scale drawing of the auditorium. The stage, which is \(35\) feet wide in real life, is \(5\) inches wide in the drawing. What is the scale of the drawing? \(1\) inch :
_
_
_
_
_
\_\_\_\_\_
_____
feet
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