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ISEE Upper Level Quantitative : How to find the volume of a cube

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #1 : Cubes

Which is the greater quantity?

(a) The volume of a cube with surface area 864 inches

(b) The volume of a cube with diagonal inches

Possible Answers:

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

The cube with the greater sidelength has the greater volume, so we need only calculate and compare sidelengths.

(a) $A=6s^{2}$ , so the sidelength of the first cube can be found as follows:

$A=6s^{2}$

$6s^{2}= 864$

$6s^{2} \div 6= 864 \div 6$

$s^{2} = 144$

$s = \sqrt{144 }= 12$ inches

(b) $d^{2} = s^{2}+ s^{2}+ s^{2} = 3 s^{2}$ by an extension of the Pythagorean Theorem, so the sidelength of the second cube can be found as follows:

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

$3 s^{2}= 21^{2}= 441$

$3 s^{2}\div 3= 441 \div 3$

$s^{2}= 147$

$s=\sqrt{ 147}$

Since 147 > 144 , $\sqrt{147 }> \sqrt{144}$ . The second cube has the greater sidelength and, subsequently, the greater volume. This makes (b) greater.

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Example Question #1 : How To Find The Volume Of A Cube

Cube 2 has twice the sidelength of Cube 1; Cube 3 has twice the sidelength of Cube 2; Cube 4 has twice the sidelength of Cube 3.

Which is the greater quantity?

(a) The mean of the volumes of Cube 1 and Cube 4

(b) The mean of the volumes of Cube 2 and Cube 3

Possible Answers:

It cannot be determined from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

The sidelengths of Cubes 1, 2, 3, and 4 can be given values s, 2s, 4s, 8s , respectively.

Then the volumes of the cubes are as follows:

Cube 1: $V= s^{3}$

Cube 2: $V= (2s)^{3} = 8s^{3}$

Cube 3: $V= (4s)^{3} = 64s^{3}$

Cube 4: $V= (8s)^{3} = 512s^{3}$

In both answer choices ask for a mean, so we can determine which answer (mean) is greater simply by comparing the sums of volumes.

(a) The sum of the volumes of Cubes 1 and 4 is $s^{3}+ 512^{3} = 513s^{3}$ .

(b) The sum of the volumes of Cubes 2 and 3 is $8s^{3}+ 64^{3} = 72s^{3}$ .

Regardless of , the sum of the volumes of Cubes 1 and 4 is greater, and therefore, so is their mean.

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Example Question #2 : Cubes

What is the volume of a cube with side length 7.236 ? Round your answer to the nearest hundredth.

Possible Answers:

314.16 in^3

123.41 in^3

452.31 in^3

612.32 in^3

378.87 in^3

Correct answer:

378.87 in^3

Explanation:

This question is relatively straightforward. The equation for the volume of a cube is:

V = s^3

(It is like doing the area of a square, then adding another dimension!)

Now, for our data, we merely need to "plug and chug:"

v=7.236^3 =378.874760256

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Example Question #3 : How To Find The Volume Of A Cube

What is the volume of a cube on which one face has a diagonal of ?

Possible Answers:

in^3

$\frac{1}{2\sqrt{2} }$ in^3

$2\sqrt{2}$ in^3

$12\sqrt{2}$ in^3

in^3

Correct answer:

$2\sqrt{2}$ in^3

Explanation:

One of the faces of the cube could be drawn like this:

Squarediagonal-2

Notice that this makes a 45-45-90 triangle.

This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both , and the hypotenuse is $\sqrt{2}$ . This will allow us to make the proportion:

$\frac{1}{\sqrt{2}} = \frac{x}{2}$

Multiplying both sides by , you get:

$x=\frac{2}{\sqrt{2}}$

Recall that the formula for the volume of a cube is:

V = s^3

Therefore, we can compute the volume using the side found above:

$V = (\frac{2}{\sqrt{2}})^3=\frac{8}{(\sqrt{2})^3}=\frac{8}{2\sqrt{2}}=\frac{4}{\sqrt2}$

Now, rationalize the denominator:

$\frac{4}{\sqrt2} * \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2}= 2\sqrt{2}$

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to find the volume of a cube

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Cubes

Example Question #1 : How To Find The Volume Of A Cube

Example Question #2 : Cubes

Example Question #3 : How To Find The Volume Of A Cube

All ISEE Upper Level Quantitative Resources

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