How to find the volume of a cube - ACT Math

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Question

The density of gold is $16\ g/cm^3$ and the density of glass is $2\ g/cm^3$ . You have a gold cube that is $x\ cm$ in length on each side. If you want to make a glass cube that is the same weight as the gold cube, how long must each side of the glass cube be?

Answer

Weight = Density * Volume

Volume of Gold Cube = side3= x3

Weight of Gold = 16 g/cm3 * x3

Weight of Glass = 3/cm3 * side3

Set the weight of the gold equal to the weight of the glass and solve for the side length:

16* x3 = 2 * side3

side3 = 16/2* x3 = 8 x3

Take the cube root of both sides:

side = 2x

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Question

What is the sum of the number of vertices, edges, and faces of a cube?

Answer

Vertices = three planes coming together at a point = 8

Edges = two planes coming together to form a line = 12

Faces = one plane as the surface of the solid = 6

Vertices + Edges + Faces = 8 + 12 + 6 = 26

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Question

What is the difference in volume between a sphere with radius x and a cube with a side of 2x? Let π = 3.14

Answer

Vcube = s3 = (2x)3 = 8x3

Vsphere = 4/3 πr3 = 4/3•3.14•x3 = 4.18x3

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Question

A tank measuring 3in wide by 5in deep is 10in tall. If there are two cubes with 2in sides in the tank, how much water is needed to fill it?

Answer

$Volume_{Tank} = Width \cdot Depth \cdot Height$

$Volume_{Tank} = 3in \cdot 5in \cdot 10in = 150 in^3$

$Volume_{Cube} = (Side)^3 = (2in)^3 = 8 in^3$

$Volume_{Water}=Volume_{Tank}-2(Volume_{Cube})$

$Volume_{Water}=150-2(8)$

$Voume_{Water}=150-16$

$Volume_{Water}=134in^3$

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Question

A cube has edges that are three times as long as those of a smaller cube. The volume of the bigger cube is how many times larger than that of the smaller cube?

Answer

If we let represent the length of an edge on the smaller cube, its volume is $x^{3}$ .

The larger cube has edges three times as long, so the length can be represented as . The volume is $(3x)^{3}$ , which is $27x^{3}$ .

The large cube's volume of $27x^{3}$ is 27 times as large as the small cube's volume of $x^{3}$ .

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Question

Chemicals to clean a swimming pool cost $0.24 per cubic foot of water. If a pool is 6 feet deep, 14 feet long and 8 feet wide, how much does it cost to clean the pool? Round to the nearest dollar.

Answer

The volume of the pool can be determined by multiplying the length, width, and height together.

$V = 6 \times 8 \times 14 = 672 ft^{3}$

Each cubit foot costs 24 cents, so:

$672ft^3\times $0.24=$161.28\approx $161$

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Question

What is the volume of a cube that has a side length of inches?

Answer

We are given the side length of a cube so we simply plug that into the formula for the volume of a cube.

That formula is , and so the correct answer is

.

Make sure to check your units, is the correct number, but the units are squared rather than cubed.

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Question

What is the volume of a cube with a side of length 1 cm?

Answer

The formula for the volume of a cube with a side of length is:
$\V = s^3 \V= 1^3 \V= 1cm^3$

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Question

A cube of sponge has volume . When water is added, the sponge triples in length along each dimension. What is the new volume of the cube, in cubic centimeters?

Answer

If our original cube has a volume of , then the length of one of its edges is $\sqrt[3]{1000} = 10cm$ . Triple each edge to , then cube the result, and we obtain

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Question

Find the volume of a cube with side length 10.

Answer

To find volume, simply cube the side length. Thus,

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Question

Find the volume of a cube whose side length is 0.1.

Answer

To solve, simply use the formula for volume of a cube. The volume of a cube is length times width times height. Since the height, width, and length are equal on a cube the formula becomes side cubed.

We are given the side length of 0.1 thus,

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Question

Find the volume of a cube given side length is .

Answer

To solve, simply cube the side length. Thus,

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Question

Find the volume of a cube given side length is .

Answer

To solve, simply cube the side length. Thus,

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Question

The density of gold is $16\ g/cm^3$ and the density of glass is $2\ g/cm^3$ . You have a gold cube that is $x\ cm$ in length on each side. If you want to make a glass cube that is the same weight as the gold cube, how long must each side of the glass cube be?

Answer

Weight = Density * Volume

Volume of Gold Cube = side3= x3

Weight of Gold = 16 g/cm3 * x3

Weight of Glass = 3/cm3 * side3

Set the weight of the gold equal to the weight of the glass and solve for the side length:

16* x3 = 2 * side3

side3 = 16/2* x3 = 8 x3

Take the cube root of both sides:

side = 2x

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Question

What is the sum of the number of vertices, edges, and faces of a cube?

Answer

Vertices = three planes coming together at a point = 8

Edges = two planes coming together to form a line = 12

Faces = one plane as the surface of the solid = 6

Vertices + Edges + Faces = 8 + 12 + 6 = 26

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Question

What is the difference in volume between a sphere with radius x and a cube with a side of 2x? Let π = 3.14

Answer

Vcube = s3 = (2x)3 = 8x3

Vsphere = 4/3 πr3 = 4/3•3.14•x3 = 4.18x3

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Question

A tank measuring 3in wide by 5in deep is 10in tall. If there are two cubes with 2in sides in the tank, how much water is needed to fill it?

Answer

$Volume_{Tank} = Width \cdot Depth \cdot Height$

$Volume_{Tank} = 3in \cdot 5in \cdot 10in = 150 in^3$

$Volume_{Cube} = (Side)^3 = (2in)^3 = 8 in^3$

$Volume_{Water}=Volume_{Tank}-2(Volume_{Cube})$

$Volume_{Water}=150-2(8)$

$Voume_{Water}=150-16$

$Volume_{Water}=134in^3$

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Question

A cube has edges that are three times as long as those of a smaller cube. The volume of the bigger cube is how many times larger than that of the smaller cube?

Answer

If we let represent the length of an edge on the smaller cube, its volume is $x^{3}$ .

The larger cube has edges three times as long, so the length can be represented as . The volume is $(3x)^{3}$ , which is $27x^{3}$ .

The large cube's volume of $27x^{3}$ is 27 times as large as the small cube's volume of $x^{3}$ .

Compare your answer with the correct one above

Question

Chemicals to clean a swimming pool cost $0.24 per cubic foot of water. If a pool is 6 feet deep, 14 feet long and 8 feet wide, how much does it cost to clean the pool? Round to the nearest dollar.

Answer

The volume of the pool can be determined by multiplying the length, width, and height together.

$V = 6 \times 8 \times 14 = 672 ft^{3}$

Each cubit foot costs 24 cents, so:

$672ft^3\times $0.24=$161.28\approx $161$

Compare your answer with the correct one above

Question

What is the volume of a cube that has a side length of inches?

Answer

We are given the side length of a cube so we simply plug that into the formula for the volume of a cube.

That formula is , and so the correct answer is

.

Make sure to check your units, is the correct number, but the units are squared rather than cubed.

Compare your answer with the correct one above

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