Resources

Resources

Volume of cubes and rectangular prisms: word problems

The formula gives the volume

V

of a rectangular pyramid with base length

l

w

h

. What is the volume, in cubic meters, of a rectangular pyramid with a base length of

5

meters, a base width of

4

meters, and a height of

3

\newline

V = \frac{1}{3} l w h

The formula gives the volume

V

of a rectangular pyramid with base length

l

w

h

. What is the volume, in cubic meters, of a rectangular pyramid with a base length of

5

meters, a base width of

4

meters, and a height of

3

\newline

V = \frac{1}{3} l w h

A top is formed by fusing two identical right circular cones by their bases as shown. The height of the top,

h

, which is the distance between the two cone points, is

6

\text{cm}

). The diameter of the circular cone bases,

d

6

cm. What is the volume of the entire object, in cubic centimeters?

If filled to capacity, what is the best estimate for the volume of the above-ground swimming pool in the picture?

\newline

5

\newline

3

\newline

30

\newline

300

\newline

3

000

\newline

30

000

10

\newline

V=\pi r^{2} h

\newline

The formula gives the volume

V

of a right circular cylinder with radius

r

h

. If the volume of a right circular cylinder with a radius of

4

feet and a height of

30

a \pi

cubic feet, what is the value of

a

The administrators of a factory modeled the cumulative average construction time in minutes per engine component as a function of the total number of components their employees had constructed,

n

, using the following function:

T(n)=3.5+7*(0.90^n)

What was the cumulative average time for an employee to construct one engine component when they first began?

\newline

3

5

\newline

7

0

\newline

10

5

\newline

14

0

A solid consists of a cone, cylinder and hemisphere joined together as shown in the diagram. Given the dimensions indicated in the diagram, find the volume and surface area of the solid, accurate to

3

significant figures.

A metal spike is made from a cube with a pyramid attached to one of the cube’s faces, as shown in the diagram. Given the dimensions indicated in the diagram, find the exact volume and exact surface area of the spike.

Ruth's grandmother gave her a beautiful hand-carved cedar chest for storing sweaters. The chest is shaped like a rectangular prism and has a volume of

12

cubic feet. It is

3

2

\newline

Which equation can you use to find how deep the chest is,

d

\newline

\newline

d = 3 \times 2 \times 12

\newline

3 \times 2 \times d = 12

\newline

How deep is the chest?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

\newline

Lucy bought a fish tank shaped like a rectangular prism for her new betta fish. The tank holds

2,244

cubic inches of water. It is

12

inches long and

11

\newline

Which equation can you use to find the height of the tank,

h

\newline

\newline

12 \times 11 \times h = 2,244

\newline

h = 12 \times 11 \times 2,244

\newline

What is the height of the tank?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

\newline

After visiting the Needleton Natural Museum, Irma was so fascinated by the reptile exhibit that she got her very own pet lizard. Irma bought a terrarium shaped like a rectangular prism for her lizard to live in. The terrarium is

20

15

inches wide, and

12

\newline

Which equation can you use to find the volume of the terrarium,

V

\newline

\newline

V = 20 \times 15 \times 12

\newline

20 \times 15 = 12 \times V

\newline

What is the volume of the terrarium?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

____ cubic inches

As a birthday gift, Lauren received a new monogrammed suitcase to take on the family trip to Ireland. The suitcase is shaped like a rectangular prism. It is

10

inches deep and

20

inches tall. The suitcase contains

2,600

cubic inches of packing space.

\newline

Which equation can you use to find the width of the suitcase,

w

\newline

\newline

10 \times w \times 20 = 2,600

\newline

10 \times 20 \times 2,600 = w

\newline

What is the width of the suitcase?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

\newline

After visiting the Needleton Natural Museum, Irma was so fascinated by the reptile exhibit that she got her very own pet lizard. Irma bought a terrarium shaped like a rectangular prism for her lizard to live in. The terrarium is

20

15

inches wide, and

12

\newline

Which equation can you use to find the volume of the terrarium,

V

\newline

\newline

20 \times 15 = 12 \times V

\newline

V = 20 \times 15 \times 12

\newline

What is the volume of the terrarium?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

____ cubic inches

The Deer Creek Aquarium has a special cube-shaped tank for its jellyfish. Each edge of the tank is

8

\newline

Which equation can you use to find the volume of the tank,

V

\newline

\newline

V = 8 + 8 + 8

\newline

V = 8 \times 8 \times 8

\newline

What is the volume of the tank?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

____ cubic meters

\newline

A rectangular prism has a volume of

693 \, \text{cubic centimeters}

. The prism has a width of

9 \, \text{centimeters}

and a height of

7 \, \text{centimeters}

. Which equation can you use to find the length of the prism,

\ell

\newline

\newline

9 \times 7 = 693 \times \ell

\newline

\ell \times 9 \times 7 = 693

\newline

What is the length of the prism?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

The Deer Creek Aquarium has a special cube-shaped tank for its jellyfish. Each edge of the tank is

8

\newline

Which equation can you use to find the volume of the tank,

V

\newline

\newline

V = 8 \times 8 \times 8

\newline

V = 8 + 8 + 8

\newline

What is the volume of the tank?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

____ cubic meters

\newline

Lucy bought a fish tank shaped like a rectangular prism for her new betta fish. The tank holds

2,244

cubic inches of water. It is

12

inches long and

11

\newline

Which equation can you use to find the height of the tank,

h

\newline

\newline

h = 12 \times 11 \times 2,244

\newline

12 \times 11 \times h = 2,244

\newline

What is the height of the tank?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

\newline

A rectangular prism has a volume of

693 \, \text{cubic centimeters}

. The prism has a width of

9 \, \text{centimeters}

and a height of

7 \, \text{centimeters}

. Which equation can you use to find the length of the prism,

\ell

\newline

\newline

\ell \times 9 \times 7 = 693

\newline

9 \times 7 = 693 \times \ell

\newline

What is the length of the prism?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

A rectangular prism has a volume of

2,730

cubic inches. The prism has a length of

15

inches and a height of

14

\newline

Which equation can you use to find the width of the prism,

w

\newline

\newline

15 \times w \times 14 = 2,730

\newline

2,730 \times 15 \times 14 = w

\newline

What is the width of the prism?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

\newline

Meg and her mom went shopping to buy Meg new pink polka-dotted winter boots. They came in a shoebox that was

14

8

inches wide, and

5

\newline

Which equation can you use to find the volume of the shoebox,

V

\newline

\newline

14 \times 8 \times 5 = V

\newline

14 = 8 \times 5 \times V

\newline

What is the volume of the shoebox?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

____ cubic inches

\newline

A rectangular prism has a volume of

2,730

cubic inches. The prism has a length of

15

inches and a height of

14

\newline

Which equation can you use to find the width of the prism,

w

\newline

\newline

2,730 \times 15 \times 14 = w

\newline

15 \times w \times 14 = 2,730

\newline

What is the width of the prism?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

\newline

Brown Deer Middle School ordered a new deep freezer to store frozen desserts in the lunchroom. The freezer is

6

2

3

\newline

Which equation can you use to find the volume of the freezer,

V

\newline

\newline

6 \times 2 \times 3 = V

\newline

6 = 2 \times 3 \times V

\newline

What is the volume of the freezer?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

____ cubic feet

\newline

As a birthday gift, Lauren received a new monogrammed suitcase to take on the family trip to Ireland. The suitcase is shaped like a rectangular prism. It is

10

inches deep and

20

inches tall. The suitcase contains

2,600

cubic inches of packing space.

\newline

Which equation can you use to find the width of the suitcase,

w

\newline

\newline

10 \times 20 \times 2,600 = w

\newline

10 \times w \times 20 = 2,600

\newline

What is the width of the suitcase?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

\newline

Meg and her mom went shopping to buy Meg new pink polka-dotted winter boots. They came in a shoebox that was

14

8

inches wide, and

5

\newline

Which equation can you use to find the volume of the shoebox,

V

\newline

\newline

14 = 8 \times 5 \times V

\newline

14 \times 8 \times 5 = V

\newline

What is the volume of the shoebox?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

____ cubic inches

Brown Deer Middle School ordered a new deep freezer to store frozen desserts in the lunchroom. The freezer is

6

2

3

\newline

Which equation can you use to find the volume of the freezer,

V

\newline

\newline

6 = 2 \times 3 \times V

\newline

6 \times 2 \times 3 = V

\newline

What is the volume of the freezer?

\newline

Write your answer as a whole number or decimal. Do not round.

\newline

____ cubic feet

\newline

The volume of a cube is

42 \frac{1}{2}

cubic centimeters. The volume of a rectangular prism is

42.7

cubic centimeters.

\newline

Which shape has a greater volume?

\newline

\newline

\newline

(B) rectangular prism

A=6s^2

The formula gives the surface area

A

of a cube with side length

8

. What is the surface area of a cube with side length of

\frac{3}{8}

Find the volume of a pyramid with a square base, where the side length of the base is

18

5

in and the height of the pyramid is

10

9

in. Round your answer to the nearest tenth of a cubic inch.

Find the volume of a regular hexagonal pyramid with side length

15

meters and a lateral edge of

17

A cylinder has a radius of

4.8

feet and a height of

8.1

feet. What is the volume, to the nearest tenth of a cubic foot, of the cylinder?

\newline

\quad989.4

\newline

\quad586.3

\newline

\quad244.3

\newline

\quad186.6

The largest U.S. state is Alaska, which covers

5.7 \times 10^5

square miles. Rhode Island, the smallest state, covers

1 \times 10^3

square miles. How many times as big as Rhode Island is Alaska? Write your answer in standard form. Do not use exponents.

\newline

____ times as big

A micron is a metric unit used to measure microscopic lengths. It is represented by the symbol

\mu m

and is equal to about

0.000039

inches. What is the approximate size of one micron written in scientific notation?

\newline

\newline

4 \times 10^{-4}

\newline

4 \times 10^{-5}

\newline

4 \times 10^{-6}

\newline

4 \times 10^{-7}

The miles of track in the U.S. railroad system peaked in the early

1900

s. At that time, more than

270,000

miles of track crisscrossed the country. What is the approximate amount of track written in scientific notation?

\newline

\newline

3 \times 10^4

\newline

3 \times 10^5

\newline

3 \times 10^6

\newline

3 \times 10^7

6 \times 10^{-2}

inches thick. The Empire State Building has a height of

1,250

1.5 \times 10^{4}

inches. How many pennies would you have to stack to reach the top of the Empire State Building?

\newline

Write your answer in scientific notation.

\newline

\newline

A typical business card is

0.013

inches thick. How would you write the approximate thickness in scientific notation?

\newline

\newline

1 \times 10^2

\newline

1 \times 10^{-2}

\newline

1 \times 10^3

\newline

1 \times 10^{-3}

Manager Mark counted the number of innings each Shire Sharks pitcher pitched during the Soiltop Tournament. Calvin pitched

11

innings, Brody pitched

5

innings, Thom pitched

12

innings, Shawn pitched

7

innings, and Harley pitched

5

\newline

Find the mean absolute deviation (MAD) of the data set.

\newline

\square

\text{innings}

A rectangular prism and cube have equal volumes. The length of the rectangular prism is

12\,\text{cm}

and its width is

8\,\text{cm}

. If each side of the cube is

12\,\text{cm}

, then what is the height of the rectangular prism in centimeters?

\newline

\square

Nora uploaded a funny video on her website, which rapidly gains views over time. The following function gives the number of views

t

days after Nora uploaded the video:

\newline

V(t)=100 \cdot e^{0.4 t}

\newline

What is the instantaneous rate of change of the number of views

4

days after the video was uploaded?

\newline

1

\newline

198

\newline

198

\newline

495

\newline

495

The side of the base of a square prism is increasing at a rate of

5

meters per second and the height of the prism is decreasing at a rate of

2

meters per second.

\newline

At a certain instant, the base's side is

6

meters and the height is

7

\newline

What is the rate of change of the volume of the prism at that instant (in cubic meters per second)?

\newline

1

\newline

348

\newline

-492

\newline

-348

\newline

492

\newline

The volume of a square prism with base side

s

h

s^{2} h

The radius of the base of a cone is increasing at a rate of

10

meters per second.

\newline

The height of the cone is fixed at

6

\newline

At a certain instant, the radius is

1

\newline

What is the rate of change of the volume of the cone at that instant (in cubic meters per second)?

\newline

1

\newline

40 \pi

\newline

200 \pi

\newline

400 \pi

\newline

20 \pi

\newline

The volume of a cone with radius

r

h

\pi r^{2} \frac{h}{3}

The surface area of a cube is increasing at a rate of

15

square meters per hour.

\newline

At a certain instant, the surface area is

24

\newline

What is the rate of change of the volume of the cube at that instant (in cubic meters per hour)?

\newline

1

\newline

8

\newline

\frac{15}{2}

\newline

(\sqrt{15})^{3}

\newline

\frac{5}{8}

The volume of a cube is increasing at a rate of

18

cubic meters per hour.

\newline

At a certain instant, the volume is

8

\newline

What is the rate of change of the surface area of the cube at that instant (in square meters per hour)?

\newline

1

\newline

(\sqrt[3]{18})^{2}

\newline

36

\newline

\frac{3}{2}

\newline

24

The side of the base of a square prism is increasing at a rate of

5

meters per second and the height of the prism is decreasing at a rate of

2

meters per second.

\newline

At a certain instant, the base's side is

6

meters and the height is

7

\newline

What is the rate of change of the volume of the prism at that instant (in cubic meters per second)?

\newline

1

\newline

492

\newline

-492

\newline

-348

\newline

348

\newline

The volume of a square prism with base side

s

h

s^{2} h

The radius of the base of a cone is increasing at a rate of

10

meters per second.

\newline

The height of the cone is fixed at

6

\newline

At a certain instant, the radius is

1

\newline

What is the rate of change of the volume of the cone at that instant (in cubic meters per second)?

\newline

1

\newline

20 \pi

\newline

40 \pi

\newline

400 \pi

\newline

200 \pi

\newline

The volume of a cone with radius

r

h

\pi r^{2} \frac{h}{3}

The surface area of a cylinder is increasing at a rate of

9 \pi

square meters per hour.

\newline

The height of the cylinder is fixed at

3

\newline

At a certain instant, the surface area is

36 \pi

\newline

What is the rate of change of the volume of the cylinder at that instant (in cubic meters per hour)?

\newline

1

\newline

9 \pi

\newline

27 \pi

\newline

\frac{\pi}{3}

\newline

\frac{1}{2}

\newline

The surface area of a cylinder with base radius

r

h

2 \pi r^{2}+2 \pi r h

\newline

The volume of a cylinder with base radius

r

h

\pi r^{2} h

A rectangular prism has base area

1804

square millimeters and height

22

millimeters. Find its volume, in cubic millimeters.

\newline

\square

A rectangular prism has a height of

11

inches and a base of

20

square inches, what is the volume of the rectangular prism?

the area of al rectangular plate is

\frac{40}{7}\,\text{m}^2

and its length is

\frac{15}{4}\,\text{m}

. find its breadth and perimeter

A building contains

24

windows. Each window measures

55 \mathrm{~cm}

95 \mathrm{~cm}

. Calculate the total area of all the windows in the building.

À La Mode Ice Cream Parlor uses the function

f(x)

to calculate its profits. When the parlor sells

x

cones, it makes

f(x)

dollars in profit.

\newline

f(50) = 85

\newline

\newline

(A) À La Mode Ice Cream Parlor makes

\$50

85

\newline

(B) À La Mode Ice Cream Parlor makes

\$85

50