GMAT Math : Calculating the volume of a cube

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Example Question #835 : Gmat Quantitative Reasoning

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

Possible Answers:

$288 \textup{ in}^{3}$

$64 \sqrt{3} \textup{ in}^{3}$

$266 \sqrt{2} \textup{ in}^{3}$

$192 \sqrt{3} \textup{ in}^{3}$

$216 \textup{ in}^{3}$

Correct answer:

$192 \sqrt{3} \textup{ in}^{3}$

Explanation:

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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Example Question #831 : Gmat Quantitative Reasoning

A cubic pool is usually filled at about $95\%$ 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?

Possible Answers:

9,500

9,500,000

950,000

950

95,000

Correct answer:

950,000

Explanation:

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

V=s^3

In our case the total volume of the pool is 10³cubic 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 = 1000 liters.

Thus,

$950 \times 1000=950,000$ .

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

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Example Question #837 : Gmat Quantitative Reasoning

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

Possible Answers:

5,832

The correct answer is not given among the other responses.

792

648

Correct answer:

5,832

Explanation:

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:

r = 9

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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Example Question #838 : Gmat Quantitative Reasoning

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

Possible Answers:

5,832

648

The correct answer is not given among the other responses.

792

Correct answer:

The correct answer is not given among the other responses.

Explanation:

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:

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

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

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

r = 9

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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Example Question #839 : Gmat Quantitative Reasoning

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?

Possible Answers:

Correct answer:

Explanation:

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:

L=2L_o=2(2)=4

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

V=L^3=(4)^3=64

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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GMAT Math : Calculating the volume of a cube

Study concepts, example questions & explanations for GMAT Math

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Example Question #835 : Gmat Quantitative Reasoning

Example Question #831 : Gmat Quantitative Reasoning

Example Question #837 : Gmat Quantitative Reasoning

Example Question #838 : Gmat Quantitative Reasoning

Example Question #839 : Gmat Quantitative Reasoning

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