How to find the volume of a cube - SSAT Upper Level Quantitative (Math)

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Question

A cube has six square faces, each with area 64 square inches. Using the conversion factor 1 inch = 2.5 centimeters, give the volume of this cube in cubic centimeters, rounding to the nearest whole number.

Answer

The volume of a cube is the cube of its sidelength, which is also the sidelength of each square face. This sidelength is the square root of the area 64:

$\sqrt{64}=8$ inches.

Multiply this by 2.5 to get the sidelength in centimeters:

$\times 2.5$ centimeters.

The cube of this is

$20^{3}$ cubic centimeters

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Question

A cube has a side length of 5 inches. Give the volume and surface area of the cube.

Answer

A cube has all edges the same length. The volume of a cube is found by multiplying the length of any edge by itself twice. As a formula:

where is the length of any edge of the cube.

The Surface Area of a cube can be calculated as .

So we get:

Volume

Surface area $=6s^2=6\times 5^2=6\times 25=150 in^2$

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Question

A cheese seller has a 2 foot x 2 foot x 2 foot block of gouda and she wants to cut it into smaller gouda cubes that are 1.5 inches on a side. How many cubes can she cut?

Answer

First we need to determine how many of the small cubes of gouda would fit along one dimension of the large cheese block. One edge of the large block is 24 inches, so 16 smaller cubes $(24\div 1.5)$ would fit along the edge. Now we simply cube this one dimension to see how many cubes fit within the whole cube. $16^{3}=4096$ .

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Question

Give the volume of a cube with surface area .

Answer

Let be the length of one edge of the cube. Since its surface area is , one face has one-sixth of this area, or $\frac{N}{6}$ . Therefore,

$s^{2} = \frac{N}{6}$ ,

and

$s =\sqrt{ \frac{N}{6}}= \frac{ \sqrt{ N}}{\sqrt{ 6}}= \frac{ \sqrt{ 6N}}{6}$

Cube this sidelength to get the volume:

$s ^{3}= \left ( \frac{ \sqrt{ 6N}}{6} \right )^{3} = \frac{\left ( \sqrt{ 6N} \right )^{3}}{6^{3}}= \frac{ 6N\sqrt{ 6N} }{216}= \frac{N\sqrt{ 6N} }{36}$

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Question

The distance from one vertex of a cube to its opposite vertex is . Give the volume of the cube.

Answer

Let be the length of one edge of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+s^{2}+s^{2}=N^{2}$

$3s^{2}=N^{2}$

$s^{2}=\frac{N^{2}}{3}$

$s =\sqrt{\frac{N^{2}}{3}} = \frac{N}{\sqrt{3}} = \frac{N\sqrt{3}}{3}$

Cube this sidelength to get the volume:

$s^{3} = \left ( \frac{N\sqrt{3}}{3} \right )^{3} = \frac{N^{3} \left ( \sqrt{3}\right ) ^{3} }{3^{3} } = \frac{N^{3} \left (3 \sqrt{3}\right ) }{27 }= \frac{N^{3} \sqrt{3} }{9 }$

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Question

The length of a diagonal of one face of a cube is 10. Give the volume of the cube.

Answer

Since a diagonal of a square face of the cube is 10, each side of each square has length $\frac{\sqrt{2}}{2}$ times this, or $10 \cdot \frac{\sqrt{2}}{2} = 5\sqrt{2}$ .

Cube this to get the volume of the cube:

$\left (5\sqrt{2} \right )^{3}= 5^{3} \left ( \sqrt{2} \right )^{3} = 125 \left (2 \sqrt{2} \right ) = 250 \sqrt{2}$ .

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Question

The distance from one vertex of a cube to its opposite vertex is one foot. Give the volume of the cube in inches.

Answer

Since we are looking at inches, we will look at one foot as twelve inches.

Let be the length of one edge of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+s^{2}+s^{2}=12^{2}$

$3s^{2}=144$

$s^{2}=48$

$s= \sqrt{48} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$ inches.

Cube this sidelength to get the volume:

$s^{3}= \left ( 4\sqrt{3} \right )^{3} = 4^{3} \cdot \left ( \sqrt{3} \right ) ^{3} = 64 \cdot 3 \sqrt{3}= 192 \sqrt{3}$ cubic inches.

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Question

An aquarium is shaped like a perfect cube; the area of each glass face is 1.44 square meters. If it is filled to the recommended 90% capacity, then, to the nearest hundred liters, how much water will it contain?

Note: 1 cubic meter = 1,000 liters.

Answer

A perfect cube has square faces; if a face has area 1.44 square meters, then each side of each face measures the square root of this, or 1.2 meters. The volume of the tank is the cube of this, or

$1.2^{3} = 1.728$ cubic meters.

Its capacity in liters is $1.728 \times 1,000 = 1,728$ liters.

90% of this is

$1,728 \times 0.9 = 1,555.2$ liters.

This rounds to 1,600 liters, the correct response.

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Question

The distance from one vertex of a perfectly cubic aquarium to its opposite vertex is meters. Give the volume of the aquarium in liters.

cubic meter = liters.

Answer

Let be the length of one edge of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+s^{2}+s^{2}=4.5^{2}$

$3s^{2}=20.25$

$s^{2} =6.75 = \frac{27}{4}$

$s = \sqrt{\frac{27}{4}} = \frac{\sqrt{27}}{\sqrt{4}}} = \frac{ \sqrt{9} \cdot \sqrt{3}}{2}= \frac{3\sqrt{3}}{2}$ meters.

Cube this sidelength to get the volume:

$s ^{3 } =\left ( \frac{3\sqrt{3}}{2} \right )^{3 }= \frac{3^{3 } \left (\sqrt{3}\right )^{3 }}{2^{3 } } = \frac{ 27 \cdot 3\sqrt{3} }{8 } = \frac{ 81 \sqrt{3} }{8 }$ cubic meters.

To convert this to liters, multiply by 1,000:

$\frac{ 81\sqrt{3} }{8 } \cdot 1,000 = \frac{81\sqrt{3} }{8 } \cdot \frac{1,000}{1} = \frac{ 81\sqrt{3} }{1 } \cdot \frac{125}{1}= 10,125\sqrt{3}$ liters.

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Question

Give the volume of a cube with surface area 3 square meters.

Answer

Let be the length of one edge of the cube. Since its surface area is 3 square meters, one face has one-sixth of this area, or $\frac{3}{6} = \frac{1}{2}$ square meters. Therefore, $s^{2} = \frac{1}{2}$ , and $s = \sqrt{ \frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}}=\frac{1 \cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}= \frac{\sqrt{2}}{2}$ meters.

The choices are in centimeters, so multiply this by 100 - the sidelength is

$\frac{\sqrt{2}}{2} \cdot 100 = 50 \sqrt{2}$ centimeters.

The volume is the cube of this, or $V= \left ( 50 \sqrt{2} \right )^{3}= 50^{3} \cdot \left ( \sqrt{2} \right )^{3}= 125,000 \cdot 2 \sqrt{2}= 250,000\sqrt{2}$ cubic centimeters.

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Question

The length of a diagonal of a cube is $15\sqrt{2}$ . Give the volume of the cube.

Answer

Let be the length of one edge of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+s^{2}+s^{2}= (15 \sqrt{2})^{2}$

$3s^{2}=15 ^{2}\cdot \left ( \sqrt{2} \right )^{2}$

$3s^{2}=225 \cdot 2 = 450$

$s^{2} = 150$

$s = \sqrt{150 } = \sqrt{25} \cdot \sqrt{6} = 5\sqrt{6}$

Cube the sidelength to get the volume:

$s ^{3 }= \left ( 5\sqrt{6} \right )^{3 }$

$s ^{3 }= 5^{3 } \left (\sqrt{6} \right )^{3 } = 125 \cdot 6 \sqrt{6} = 750 \sqrt{6}$

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Question

The length of a diagonal of one face of a cube is $16\sqrt{2}$ . Give the volume of the cube.

Answer

A diagonal of a square has length $\sqrt{2}$ times that of a side, so each side of each square face of the cube has length $16 \sqrt{2} \div \sqrt{2}= 16$ . Cube this to get the volume:

$16^{3} = 4,096$

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Question

Find the volume of a cube with a side length of .

Answer

Write the formula for the volume of a cube.

$V=a^3=(20)^3=8\textup{,}000$

The correct answer is $8\textup{,}000$ .

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