Calculating the volume of a cube - GMAT Quantitative Reasoning

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Question

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

Answer

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Question

The length, width, and height of a rectangular prism, in inches, are three different prime numbers. All three dimensions are between six feet and seven feet. What is the volume of the prism?

Answer

Six feet and seven feet are equal to, respectively, 72 inches and 84 inches. There are three different prime numbers between 72 and 84 - 73, 79, and 83 - so these are the three dimensions of the prism in inches. The volume of the prism is

$73 \times 79 \times 83 = 478,661$ cubic inches.

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Question

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

Answer

Since we are looking at yards, we will look at twelve feet as four yards.

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

$s^{2}+s^{2}+s^{2}=4^{2}$

$3s^{2}=16$

$s^{2}= \frac{16}{3}$

$s^{2}= \sqrt{\frac{16}{3}} = \frac{\sqrt{16}}{\sqrt{3}} = \frac{4}{\sqrt{3}}$ yards.

Cube this sidelength to get the volume:

$s^{3}= \left ( \frac{4}{\sqrt{3}} \right )^{3} = \frac{4^{3}}{(\sqrt{3})^{3}} = \frac{64 }{3\sqrt{3} }= \frac{64 \cdot \sqrt{3}}{3\sqrt{3}\cdot \sqrt{3} } = \frac{64 \sqrt{3}}{9}$ cubic yards.

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Question

The length of a diagonal of one face of a cube is $5\sqrt{2}$ . 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 $5 \sqrt{2} \div \sqrt{2}= 5$ . Cube this to get the volume:

$V= 5^{3} = 125$

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Question

The length of a diagonal of a cube is $\sqrt{17}$ . 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,

$s^{2}+s^{2}+s^{2}= ( \sqrt{17})^{2}$

$3s^{2}= 17$

$s^{2}= \frac{17}{3}$

$s =\sqrt{ \frac{17}{3}}$

Cube the sidelength to get the volume:

$s^{3} =\left (\sqrt{ \frac{17}{3}} \right )^{3}$

$=\frac{ \left (\sqrt{ 17} \right )^{3}}{\left (\sqrt{3} \right )^{3}} \right )$

$=\frac{ 17 \sqrt{ 17} }{ 3 \sqrt{3} } \right )$

$=\frac{ 17 \sqrt{ 17} \cdot \sqrt{3}}{ 3 \sqrt{3} \cdot \sqrt{3}} \right )$

$=\frac{ 17 \sqrt{ 51} }{ 9} \right )$

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Question

Which choice comes closest to the volume of a cube with surface area square centimeters?

Answer

The suface area of a cube is six times the square of the length of one side, so solve for in the following:

$6s^{2} =135,000$

$s^{2} = 22,500$

This is the sidelength in centimeters; since we are looking at meters, divide this by 100 to convert to $150 \div 100 = 1.5$ meters.

Cube this to get volume

$V =1.5^{3}= 3.375$ cubic meters.

Of the given choices, 3.5 cubic meters comes closest.

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Question

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

Note: cubic meter liters.

Answer

A perfect cube has square faces; if a face has area 6.25 square meters, then each side of each face measures the square root of this, or 2.5 meters. The volume of the tank is the cube of this, or

$2.5^{3} = 15.625$ cubic meters.

Its capacity in liters is $15.625 \times 1,000 = 15,625$ liters.

80% of this is

$15,625 \times 0.8 = 12,500$ liters.

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Question

A sphere with surface area $40 \pi$ is inscribed inside a cube. Give the volume of the cube.

Answer

The sidelength of the cube is the diameter of the inscribed sphere, which is twice that sphere's radius. The sphere has surface area $40 \pi$ , so the radius is calculated as follows:

$4 \pi r^{2} = SA$

$4 \pi r^{2} = 40 \pi$

$r^{2} = 40 \pi \div 4 \pi = 10$

$r = \sqrt{10}$

The diameter of the sphere - and the sidelength of the cube - is twice this, or $2 \sqrt{10}$ .

Cube this sidelength to get the volume of the cube:

$V= \left ( 2 \sqrt{10} \right )^{3} = 2^{3} \cdot\left ( \sqrt{10 } \right )^{3} = 8 \cdot 10 \sqrt{10} = 80 \sqrt{10}$

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Question

A sphere with volume $36 \pi$ is inscribed inside a cube. Give the volume of the cube.

Answer

The sidelength of the cube is the diameter of the inscribed sphere, which is twice that sphere's radius. The sphere has volume $36 \pi$ , so the radius is calculated as follows:

$\frac{4}{3} \pi r^{3} = V$

$\frac{4}{3} \pi r^{3} = 36 \pi$

$r^{3} = 36\pi \div \left (\frac{4}{3} \pi \right )$

$r^{3} = 27$

$r = \sqrt[3]{27} = 3$

The diameter of the sphere - and the sidelength of the cube - is twice this, or 6. Cube this sidelength to get the volume of the cube:

$V = 6^{3} = 216$

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Question

A cube is inscribed inside a sphere with volume $36 \pi$ . Give the volume of the cube.

Answer

The diameter of the circle - twice its radius - coincides with the length of a diagonal of the inscribed cube. The sphere has volume $36 \pi$ , so the radius is calculated as follows:

$\frac{4}{3} \pi r^{3} = V$

$\frac{4}{3} \pi r^{3} = 36 \pi$

$r^{3} = 36\pi \div \left (\frac{4}{3} \pi \right )$

$r^{3} = 27$

$r = \sqrt[3]{27} = 3$

The diameter of the sphere - and the length of a diagonal of the cube - is twice this, or 6.

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

$s^{2}+s^{2}+s^{2}=6^{2}$

$3s^{2}=36$

$s^{2}=12$

$s = \sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}$

The volume of the cube is the cube of this, or

$V=\left (2 \sqrt{3} \right )^{3} =2^{3} \left ( \sqrt{3} \right )^{3} = 8 \cdot 3 \sqrt{3} = 24 \sqrt{3}$

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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.

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,

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

$3s^{2}=144$

$s^{2}=48$

$s= \sqrt{48} = 4\sqrt{3}$ 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

A cubic pool is usually filled at about of its total volume. If one side of the pool is m, how many liters of water will fill the pool to the desired capacity?

Answer

To find the volume of a cube we use the formula:

In our case the total volume of the pool is 103 cubic meters.

Filling the pool to 95% of its total volume will require: $0.95\times1000 = 950$ cubic meters.

Now we need to convert from cubic meters to liters in order to answer the question.

Remember cubic meter = liters.

Thus,

$950 \times 1000=950,000$ .

Therefore, the number of liters of water needed is 950000 liters.

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Question

A sphere with surface area $324 \pi$ circumscribes a cube. Give the volume of the cube.

Answer

The diameter of a sphere is equal to the length of a diagonal of the cube it circumscribes. We can derive the radius using the formula for the volume of a sphere, but first, we have to solve for the radius using the formula for the surface area of a sphere:

$SA = 4 \pi r^{2}$

$4 \pi r^{2} = 324 \pi$

$r^{2} = \frac{324 \pi}{4 \pi }= 81$

The diameter is twice this, or . This is the length of the diagonal of the cube. By the three-dimensional extension of the Pythagorean Theorem,

$s^{2}+ s^{2}+ s^{2} = 18^{2}$

$3s^{2} = 324$

$s^{2} =108$

$s = \sqrt{108} = \sqrt{36} \cdot \sqrt{3} = 6 \sqrt{3}$

The volume is the cube of this:

$V =(6 \sqrt{3}) ^{3}= 6^{3} \cdot \left (\sqrt{3} \right )^{3} = 216 \cdot 3 \cdot \sqrt{3} = 648 \sqrt{3}$ , which is not given as one of the choices.

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Question

A sphere of surface area $324 \pi$ is inscribed inside a cube. Give the volume of the cube.

Answer

The diameter of a sphere is equal to the length of an edge of the cube in which it is inscribed. We can derive the radius using the formula for the surface area of a sphere:

$SA = 4 \pi r^{2}$

Substituting in the given value:

$4 \pi r^{2} = 324 \pi$

Dividing each side by $4\pi$ :

$r^{2} = \frac{324 \pi}{4 \pi }= 81$

Taking the square root of each side:

The diameter is twice this, or . This is the length of the edge of the cube, so we can cube it to get the volume:

$V =18 ^{3}= 5,832$

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Question

A cube has a volume of cubic feet. If the length of each side of the cube is doubled, what is its new volume, in cubic feet?

Answer

One way we can solve this problem is by first determining the dimensions of the original cube. By definition, the length, width, and depth of a cube are equal, so if the original volume is 8 cubic feet, then:

$V_o=L_o^3=8\rightarrow L_o=\sqrt[3]{8}=2$

We then double the length of each side of the original cube, so the length of each side of the new cube is:

Now we can use the dimensions of our new cube to find its volume:

So if we double the dimensions of the original cube, the resulting cube has a volume that is eight times greater, 64 cubic feet.

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