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Math Problems
Geometry
Pythagorean theorem
What is the length of the diagonal,
d
d
d
, of the rectangular prism shown below? Round your answer to the nearest tenth.
\newline
A rectangular prism that has a length of
8
8
8
units, a width of
2
2
2
units, and a height of
5
5
5
units. Diagonal
D
D
D
starts at the bottom left corner of the front face of the prism and ends at the top left corner of the back face. The length is units.
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From a hot-air balloon, Justin measures a
3
4
degrees
angle of depression
34^\text{degrees} \text{ } \text{angle of depression}
3
4
degrees
angle of depression
to a landmark that’s
1424
feet
1424 \text{ feet}
1424
feet
away, measuring horizontally. What’s the balloon’s vertical distance above the ground? Round your answer to the nearest hundredth of a foot if necessary.
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Measure the time it takes you to walk from your house to your nearest shopping center If you consider that your average walking speed is
4
km/h
4\,\text{km/h}
4
km/h
, then estimate the distance of the shopping center from your house.
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A boat is heading towards a lighthouse, whose beacon-light is
135
135
135
feet above the water. The boat’s crew measures the angle of elevation to the beacon,
4
∘
4^\circ
4
∘
. What is the ship’s horizontal distance from the lighthouse (and the shore)? Round your answer to the nearest hundredth of a foot if necessary.
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You are standing
42
42
42
feet away from the base of the Statue of Liberty looking up at an angle of
82.
2
∘
82.2^\circ
82.
2
∘
. How tall is the statue?
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A
35
35
35
foot ladder is set against the side of a house so that it reaches up
21
21
21
feet. If Elijah grabs the ladder at its base and pulls it
4
4
4
feet farther from the house, how far up the side of the house will the ladder reach now?
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While painting the side of his house, Jason positioned his ladder so that the top was
16
16
16
feet above the ground, while the bottom was
12
12
12
feet away from the base of the wall. To reach a spot higher up on the house, Jason moved the ladder so that the top was
18
18
18
feet above the ground. How much closer to the house is the bottom of the ladder now? If necessary, round your answer to the nearest tenth.
\newline
____ feet
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A chessboard is one large square made up of
8
8
8
rows of
8
8
8
equal sized smaller squares. Krysta has two chessboards. On one board, the small squares are
2
2
2
inches by
2
2
2
inches. On the other, they are
3
3
3
inches by
3
3
3
inches. How much longer is the diagonal of the larger board than the diagonal of the smaller board? If necessary, round your answer to the nearest tenth.
\newline
____ inches
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How long is the shortest route from Mesa Vista to Gibbs using the roads shown?
\newline
\newline
If necessary, round your answer to the nearest tenth.
\newline
\newline
_
_
_
_
\_\_\_\_
____
miles
\newline
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In Rosewood City, the library is
6.5
6.5
6.5
miles due south of the courthouse and
3.4
3.4
3.4
miles due west of the community swimming pool. What is the distance between the courthouse and the community swimming pool?
\newline
If necessary, round your answer to the nearest tenth.
\newline
____ miles
\newline
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Chad is decorating a rectangular ballroom ceiling with decorative lights. Beginning in a corner, Chad strings the lights along the length of the ceiling, a distance of
55
55
55
feet. Next, Chad strings the lights along the width of the ceiling, a distance of
48
48
48
feet. Then Chad strings the lights straight back to the original corner. At this point, how many feet of lights has Chad used?
\newline
If necessary, round your answer to the nearest tenth.
\newline
____ feet
\newline
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Rachel is sitting in a tree
15
15
15
feet above the ground. A minute ago, she spotted a skunk
40
40
40
feet from the base of the tree walking closer to her. Now, the skunk is
20
20
20
feet from the base of the tree. The diagonal line from Rachel to the skunk is the distance between them. How much closer is the skunk to Rachel now than it was a minute ago?
\newline
If necessary, round your answer to the nearest tenth.
\newline
_
_
_
_
\_\_\_\_
____
feet
\newline
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Brenna is going to swim the length of her rectangular swimming pool. The width of the pool is
36
36
36
feet, and the diagonal distance from one corner to the opposite corner is
60
60
60
feet. What is the length of the pool?
\newline
If necessary, round your answer to the nearest tenth.
\newline
\newline
_
_
_
_
\_\_\_\_
____
feet
\newline
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The Island Loop Ferry takes the same route multiple times a day. It starts at the port of Seabreeze Village and travels due south for
8
8
8
miles to the town of Oceanview. From there, it travels due east for
15
15
15
miles to Portlandville. Finally, it makes a straight-line journey back to Seabreeze Village. If the ferry can travel
5
5
5
miles on a gallon of fuel, how many gallons does the ferry use to complete the route one time? If necessary, round your answer to the nearest tenth.
\newline
____ gallons
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Rose is building a slide for her kids. The ladder is
8
8
8
feet tall and the slide is
17
17
17
feet long. What is the distance between the ladder and the bottom of the slide?
\newline
_
_
_
_
\_\_\_\_
____
feet
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How long is the shortest route from McLain to Austinburg using the roads shown?
\newline
\newline
If necessary, round your answer to the nearest tenth.
\newline
\newline
_
_
_
_
\_\_\_\_
____
miles
\newline
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Tim likes to skate at a candy store that is due south of his school and due west of his favorite ice cream parlor. If the candy store is \(12\) kilometers from his school and the straight-line distance between the school and the ice cream parlor is \(13\) kilometers, how far is the candy store from the ice cream parlor?
\newline
____________Kilometers
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A water halloon is being launched into the air at a
7
3
∘
73^{\circ}
7
3
∘
angle of elevation from the ground. The water balloon reaches a height of
30
30
30
feet directly above your head. How far from you was the water balloon launched?
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Isabella is playing with her yo-yo. The vertical distance
Y
Y
Y
(in
c
m
\mathrm{cm}
cm
) between the yo-yo and her hand
t
t
t
seconds after she first spins it out is modeled by the following function. Here,
t
t
t
is in radians.
\newline
Y
(
t
)
=
40
cos
(
2
π
3
t
)
−
71
Y(t)=40 \cos \left(\frac{2 \pi}{3} t\right)-71
Y
(
t
)
=
40
cos
(
3
2
π
t
)
−
71
\newline
How long does it take the yo-yo to fall all the way down from its peak, and then rise up to a vertical distance of
−
80
c
m
-80 \mathrm{~cm}
−
80
cm
?
\newline
Round your final answer to the nearest tenth of a second.
\newline
□
\square
□
seconds
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Bilal's fishing line was on a spool with a radius of
4
c
m
4 \mathrm{~cm}
4
cm
. Suddenly, a fish pulled on the line, and the spool spun
16
16
16
times before Bilal began to reel in the fish.
\newline
What is the distance the fish pulled the fishing line?
\newline
Round your answer to the nearest
c
m
\mathrm{cm}
cm
.
\newline
□
\square
□
c
m
\mathrm{cm}
cm
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You are going on a field trip in a school bus. A special detector placed on one of the bus's tires counts the number of rotations completed by the tire during the journey. The tire's diameter is
1
1
1
meter, and it made
8000
8000
8000
rotations during the trip.
\newline
How far did the bus travel?
\newline
Round your answer to the nearest meter.
\newline
□
\square
□
meters
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The second hand on a clock is
8
c
m
8 \mathrm{~cm}
8
cm
long.
\newline
What is the distance the tip of the second hand travels in
10
10
10
minutes?
\newline
Round your answer to the nearest
c
m
\mathrm{cm}
cm
.
\newline
□
\square
□
c
m
\mathrm{cm}
cm
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Aliaa is skating down a ramp. The wheels on her skateboard have diameters of
60
m
m
60 \mathrm{~mm}
60
mm
.
\newline
What is the distance Aliaa travels when her skateboard wheels make
20
20
20
revolutions?
\newline
Round your answer to the nearest
m
m
\mathrm{mm}
mm
.
\newline
□
\square
□
m
m
\mathrm{mm}
mm
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A wet bicycle tire leaves a trace of water on the floor. The tire has a radius of
30
c
m
30 \mathrm{~cm}
30
cm
, and the bicycle wheel makes
3
3
3
full rotations before stopping.
\newline
How long is the trace of water left on the floor?
\newline
Round your answer to the nearest
c
m
\mathrm{cm}
cm
.
\newline
□
\square
□
cm
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A gopher has dug holes in opposite corners of a rectangular yard. One length of the yard is
10
meters
10\,\text{meters}
10
meters
and the distance between the gopher's holes is
26
meters
26\,\text{meters}
26
meters
. How wide is the yard?
\newline
_
_
_
_
_
\_\_\_\_\_
_____
meters
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Beth ran the width of a field, a distance of
35
35
35
yards. Then she ran the length of the field, a distance of
84
84
84
yards. Finally, she ran diagonally across it to get back to her starting point. How far did Beth run?
\newline
_
_
_
_
\_\_\_\_
____
yards
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On the school playground, the slide is due west of the tire swing and due south of the monkey bars. If the distance between the slide and the tire swing is
40
40
40
feet and the distance between the tire swing and the monkey bars is
50
50
50
feet, how far is the slide from the monkey bars?
\newline
_
_
_
_
_
_
_
\_\_\_\_\_\_\_
_______
feet
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Julian is creating a stained glass windowpane that will be
72
72
72
centimeters by
21
21
21
centimeters. To make the rectangular windowpane, Julian fuses together two triangles of colored glass. What is the perimeter of one triangle?
\newline
_____ centimeters
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A mouse has made holes in opposite corners of a rectangular kitchen. Starting from its hole in the northwest corner, the mouse scurries
20
20
20
feet along the length of the kitchen to reach a piece of cheese in the southwest corner. Then the mouse scurries
15
15
15
feet along the width of the kitchen to its other hole in the southeast corner. Finally the mouse scurries back to the first hole. What is the total distance the mouse scurries?
\newline
_____ feet
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Patty climbs a
40
40
40
-foot tower to ride a straight water slide. After landing in the splash pool, she walks
42
42
42
feet back to the base of the tower. How far did she travel in all?
\newline
_
_
_
_
\_\_\_\_
____
feet
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Damon is playing on the slide. Over and over, he climbs the
8
8
8
-foot ladder, goes down the slide, and walks
6
6
6
feet to get back to the ladder. How far does Damon travel each time he repeats this process?
\newline
_____ feet
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Polly measured the floor of her storage unit, which is rectangular. It is
14
14
14
feet wide and
50
50
50
feet from one corner to the opposite corner. How long is the storage unit?
\newline
_
_
_
_
\_\_\_\_
____
feet
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An envelope measures
9
9
9
centimeters by
40
40
40
centimeters. A pencil is placed in the envelope at a diagonal. What is the maximum possible length of the pencil?
\newline
_____ centimeters
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Three ballet dancers are positioned on stage. Ruth is
15
15
15
feet straight behind Pablo and
20
20
20
feet directly left of Chase. When the music begins, Ruth twirls to Chase's position, then leaps to Pablo's position, and finally walks back to her original position. How far did Ruth travel?
\newline
_____ feet
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Sidney is deep-sea diving with Aiden, who is to the south of her. A fish emerges from a hole to the east of Aiden. The fish swims
24
meters
24\,\text{meters}
24
meters
to Aiden and then swims
10
meters
10\,\text{meters}
10
meters
to Sidney. The fish then swims back to its hole and disappears. How far did the fish travel in all?
\newline
_
_
_
_
_
\_\_\_\_\_
_____
meters
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A jogger running around a rectangular park takes a shortcut back to her car by jogging from one corner of the park to the diagonally opposite corner. If the park is
60
yards
60\,\text{yards}
60
yards
by
45
yards
45\,\text{yards}
45
yards
, how far is the shortcut?
\newline
_
_
_
_
_
\_\_\_\_\_
_____
yards
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Three baseball players are playing catch. Eli is
15
15
15
feet south of Alexandra and
20
20
20
feet west of Matt. How far does Alexandra need to throw the ball to get it to Matt?
\newline
_
_
_
_
\_\_\_\_
____
feet
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Alexandra is going to swim the length of her swimming pool. The width of her pool is
21
21
21
meters and the diagonal distance from one corner to the opposite corner is
29
29
29
meters. How far will Alexandra swim when she swims the length?
\newline
_____ meters
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Three students are sitting on a school bus. Billy is
4
meters
4\,\text{meters}
4
meters
directly behind Sidney and
3
meters
3\,\text{meters}
3
meters
directly left of Gordon. Billy makes a paper airplane and throws it to Sidney. Sidney throws the airplane to Gordon, who throws it back to Billy. How far has the paper airplane traveled?
\newline
_
_
_
_
_
\_\_\_\_\_
_____
meters
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If you place a
24
24
24
-foot ladder against the top of a
20
20
20
-foot building, how many feet will the bottom of the ladder be from the bottom of the building? Round to the nearest tenth of a foot.
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A frog is in a
40
m
40\,\text{m}
40
m
deep place. Each day, the frog climbs
3
m
3\,\text{m}
3
m
, but falls back
1
m
1\,\text{m}
1
m
at night. How many days will it take to get out?
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Roselyn is driving to visit her family, who live
150
150
150
kilometers away. Her average speed is
60
60
60
kilometers per hour. The car's tank has
20
20
20
liters of fuel at the beginning of the drive, and its fuel efficiency is
6
6
6
kilometers per liter. Fuel costs
0.60
0.60
0.60
dollars per liter. How long can Roselyn drive before she runs out of fuel?
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You are on the roof of a building and see a friend standing
190
190
190
meters away from the base of a building. The angle of depression to your friend is
2
8
∘
28^{\circ}
2
8
∘
. How far below are they?
\newline
They are
□
\square
□
meters below you (round your answer to three decimal places)
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In Belleville, the library is due south of the courthouse and due west of the community swimming pool. If the distance between the library and the courthouse is
6.6
6.6
6.6
kilometres and the distance between the courthouse and the city pool is
8.2
8.2
8.2
kilometres, how far is the library from the community pool? If necessary, round to the nearest tenth.
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An electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box
22
22
22
feet up. The ladder makes an angle of
6
6
∘
66^\circ
6
6
∘
with the ground. Find the length of the ladder. Round your answer to the nearest tenth of a foot if necessary.
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Alice is driving north at
80
k
m
/
h
80 \mathrm{~km} / \mathrm{h}
80
km
/
h
and Bob is driving east at
50
k
m
/
h
50 \mathrm{~km} / \mathrm{h}
50
km
/
h
. Both cars are approaching the intersection of the two roads. At what rate is the distance between the two cars decreasing when Alice's car is
0.5
k
m
0.5 \mathrm{~km}
0.5
km
from the intersection and Bob's car is
0.8
k
m
0.8 \mathrm{~km}
0.8
km
from the intersection?
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A car travels at
54
km/h
54\,\text{km/h}
54
km/h
for first
20
s
20\,\text{s}
20
s
,
36
km/h
36\,\text{km/h}
36
km/h
for next
30
s
30\,\text{s}
30
s
and finally
18
km/h
18\,\text{km/h}
18
km/h
for next
10
s
10\,\text{s}
10
s
. Find its average speed.
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Luis is flying a kite, holding his hands a distance of
3.75
3.75
3.75
feet above the ground and letting all the kite’s string play out. He measures the angle of elevation from his hand to the kite to be
3
5
∘
35^\circ
3
5
∘
. If the string from the kite to his hand is
140
140
140
feet long, how many feet is the kite above the ground? Round your answer to the nearest tenth of a foot if necessary.
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A
20
20
20
-foot fall flagpole is leaning towards a house. If the flagpole makes an
8
5
∘
85^\circ
8
5
∘
angle with the ground and the angle of elevation from the base of the house to the top of the pole is
55
55
55
degrees, find the distance from the base of the flagpole to the house.
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Chelsea is sitting
8
8
8
feet from the foot of a tree. From where she is sitting, the angle of elevation of her line of sight to the top of the tree is
3
6
∘
36^{\circ}
3
6
∘
. If her line of sight starts
1.5
1.5
1.5
feet above ground, how tall is the tree, to the nearest foot?
\newline
(1)
8
8
8
\newline
(2)
7
7
7
\newline
(3)
6
6
6
\newline
(4)
4
4
4
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2
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