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ISEE Upper Level Quantitative : Cubes

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

$A=6s^{2}$ $\displaystyle A=6s^{2}$

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

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

$s^{2} = 144$ $\displaystyle s^{2} = 144$

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

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

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

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

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

$s^{2}= 147$ $\displaystyle s^{2}= 147$

$s=\sqrt{ 147}$ $\displaystyle s=\sqrt{ 147}$

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

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Example Question #1 : Cubes

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:

(a) is greater.

It cannot be determined from the information given.

(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 $\displaystyle s, 2s, 4s, 8s$, respectively.

Then the volumes of the cubes are as follows:

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

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

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

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

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 $s^{3}+ 512^{3} = 513s^{3}$ $\displaystyle s^{3}+ 512^{3} = 513s^{3}$.

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

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

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Example Question #1 : Cubes

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

Possible Answers:

123.41 $\displaystyle 123.41$ in^3 $\displaystyle in^3$

378.87 $\displaystyle 378.87$ in^3 $\displaystyle in^3$

612.32 $\displaystyle 612.32$ in^3 $\displaystyle in^3$

314.16 $\displaystyle 314.16$ in^3 $\displaystyle in^3$

452.31 $\displaystyle 452.31$ in^3 $\displaystyle in^3$

Correct answer:

378.87 $\displaystyle 378.87$ in^3 $\displaystyle in^3$

Explanation:

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

V = s^3 $\displaystyle V = s^3$

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

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

$\displaystyle v=7.236^3 =378.874760256$

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Example Question #13 : Solid Geometry

What is the volume of a cube on which one face has a diagonal of $\displaystyle 2$ $\displaystyle in$?

Possible Answers:

$\displaystyle 12$ in^3 $\displaystyle in^3$

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

$12\sqrt{2}$ $\displaystyle 12\sqrt{2}$ in^3 $\displaystyle in^3$

$2\sqrt{2}$ $\displaystyle 2\sqrt{2}$ in^3 $\displaystyle in^3$

$\displaystyle 2$ in^3 $\displaystyle in^3$

Correct answer:

$2\sqrt{2}$ $\displaystyle 2\sqrt{2}$ in^3 $\displaystyle in^3$

Explanation:

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

Squarediagonal-2

Notice that this makes a $\displaystyle 45-45-90$ triangle.

This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both $\displaystyle 1$, and the hypotenuse is $\sqrt{2}$ $\displaystyle \sqrt{2}$. This will allow us to make the proportion:

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

Multiplying both sides by $\displaystyle 2$, you get:

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

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

V = s^3 $\displaystyle V = s^3$

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

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

Now, rationalize the denominator:

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

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

The volume of a cube is 343 cubic inches. Give its surface area.

Possible Answers:

$343 \; \textrm{in}^{2}$ $\displaystyle 343 \; \textrm{in}^{2}$

$294 \; \textrm{in}^{2}$ $\displaystyle 294 \; \textrm{in}^{2}$

$196 \; \textrm{in}^{2}$ $\displaystyle 196 \; \textrm{in}^{2}$

$392 \; \textrm{in}^{2}$ $\displaystyle 392 \; \textrm{in}^{2}$

$242\; \textrm{in}^{2}$ $\displaystyle 242\; \textrm{in}^{2}$

Correct answer:

$294 \; \textrm{in}^{2}$ $\displaystyle 294 \; \textrm{in}^{2}$

Explanation:

The volume of a cube is defined by the formula

$V=s^{3}$ $\displaystyle V=s^{3}$

where $\displaystyle s$ is the length of one side.

If V=343 $\displaystyle V=343$, then

$s^{3} = 343$ $\displaystyle s^{3} = 343$

and

$s = \sqrt[3]{343} = 7$ $\displaystyle s = \sqrt[3]{343} = 7$

So one side measures 7 inches.

The surface area of a cube is defined by the formula

$A = 6s^{2}$ $\displaystyle A = 6s^{2}$ , so

$A = 6 \cdot 7^{2} = 294$ $\displaystyle A = 6 \cdot 7^{2} = 294$

The surface area is 294 square inches.

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Example Question #5 : Cubes

What is the surface area of a cube with side length 5.7 $\displaystyle 5.7$ $\displaystyle in$?

Possible Answers:

194.94 $\displaystyle 194.94$ in^2 $\displaystyle in^2$

32.49 $\displaystyle 32.49$ in^2 $\displaystyle in^2$

185.193 $\displaystyle 185.193$ in^2 $\displaystyle in^2$

91.18 $\displaystyle 91.18$ in^2 $\displaystyle in^2$

173.47 $\displaystyle 173.47$ in^2 $\displaystyle in^2$

Correct answer:

194.94 $\displaystyle 194.94$ in^2 $\displaystyle in^2$

Explanation:

Recall that the formula for the surface area of a cube is:

$\displaystyle SA = 6 * s^2$, where $\displaystyle s$ is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side ( s^2 $\displaystyle s^2$) by $\displaystyle 6$ because a cube has $\displaystyle 6$ equal sides.

For our data, we know that s = 5.7 $\displaystyle s = 5.7$; therefore, our equation is:

$\displaystyle SA = 6 * 5.7^2 = 6 * 32.49 = 194.94$

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

What is the surface area of a cube with a volume 512 $\displaystyle 512$ in^3 $\displaystyle in^3$?

Possible Answers:

384 $\displaystyle 384$ in^2 $\displaystyle in^2$

193.44 $\displaystyle 193.44$ in^2 $\displaystyle in^2$

192 $\displaystyle 192$ in^2 $\displaystyle in^2$

321 $\displaystyle 321$ in^2 $\displaystyle in^2$

85.33 $\displaystyle 85.33$ in^2 $\displaystyle in^2$

Correct answer:

384 $\displaystyle 384$ in^2 $\displaystyle in^2$

Explanation:

To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:

V = s^3 $\displaystyle V = s^3$, where $\displaystyle s$ is the side length.

For our data, this gives us:

$\displaystyle s^3 = 512$

Taking the cube-root of both sides, we get:

s = 8 $\displaystyle s = 8$

Now, use the surface area formula to compute the total surface area:

For our data, this gives us:

$\displaystyle SA = 6*8^2 = 6*64=384$

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Example Question #1 : Cubes

What is the surface area of a cube with a volume 274.625 $\displaystyle 274.625$ in^3 $\displaystyle in^3$?

Possible Answers:

126.75 $\displaystyle 126.75$ in^2 $\displaystyle in^2$

344.13 $\displaystyle 344.13$ in^2 $\displaystyle in^2$

549.25 $\displaystyle 549.25$ in^2 $\displaystyle in^2$

253.5 $\displaystyle 253.5$ in^2 $\displaystyle in^2$

631.13 $\displaystyle 631.13$ in^2 $\displaystyle in^2$

Correct answer:

253.5 $\displaystyle 253.5$ in^2 $\displaystyle in^2$

Explanation:

To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:

V = s^3 $\displaystyle V = s^3$, where $\displaystyle s$ is the side length.

For our data, this gives us:

$\displaystyle s^3 = 274.625$

Taking the cube-root of both sides, we get:

s = 6.5 $\displaystyle s = 6.5$

(You will need to use a calculator for this. If your calculator gives you something like $\displaystyle 6.599999999999$ . . . it is okay to round. This is just the nature of taking roots!).

Now, use the surface area formula to compute the total surface area:

For our data, this gives us:

$\displaystyle SA = 6*6.5^2 = 6*42.25=253.5$

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Example Question #5 : How To Find The Surface Area Of A Cube

What is the surface area for a cube with a diagonal length of $3\sqrt{3}$ $\displaystyle 3\sqrt{3}$ $\displaystyle in$?

Possible Answers:

$28\sqrt{3}$ $\displaystyle 28\sqrt{3}$

$15\sqrt{6}$ $\displaystyle 15\sqrt{6}$

$\displaystyle 27$

$3\sqrt{3}$ $\displaystyle 3\sqrt{3}$

$\displaystyle 54$

Correct answer:

$\displaystyle 54$

Explanation:

Now, this could look like a difficult problem; however, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:

$D = \sqrt{3s^2}$ $\displaystyle D = \sqrt{3s^2}$

(It is very easy, because the three lengths are all the same: $\displaystyle s$).

So, we know this, then:

$3\sqrt{3}=\sqrt{3s^2}$ $\displaystyle 3\sqrt{3}=\sqrt{3s^2}$

To solve, you can factor out an $\displaystyle s$ from the root on the right side of the equation:

$3\sqrt{3}=s\sqrt{3}$ $\displaystyle 3\sqrt{3}=s\sqrt{3}$

Just by looking at this, you can tell that the answer is:

s=3 $\displaystyle s=3$

Now, use the surface area formula to compute the total surface area:

For our data, this is:

$\displaystyle SA = 6 * 3^2 = 6*9=54$

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Example Question #6 : How To Find The Surface Area Of A Cube

What is the volume of a cube with a diagonal length of $7\sqrt{3}$ $\displaystyle 7\sqrt{3}$ $\displaystyle in$?

Possible Answers:

343 $\displaystyle 343$ in^3 $\displaystyle in^3$

$1029\sqrt{3}$ $\displaystyle 1029\sqrt{3}$ in^3 $\displaystyle in^3$

$14\sqrt{3}$ $\displaystyle 14\sqrt{3}$ in^3 $\displaystyle in^3$

$343\sqrt{3}$ $\displaystyle 343\sqrt{3}$ in^3 $\displaystyle in^3$

171.5 $\displaystyle 171.5$ in^3 $\displaystyle in^3$

Correct answer:

343 $\displaystyle 343$ in^3 $\displaystyle in^3$

Explanation:

Now, this could look like a difficult problem. However, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:

$D = \sqrt{3s^2}$ $\displaystyle D = \sqrt{3s^2}$

(It is very easy, because the three lengths are all the same: $\displaystyle s$).

So, we know this, then:

$7\sqrt{3}=\sqrt{3s^2}$ $\displaystyle 7\sqrt{3}=\sqrt{3s^2}$

To solve, you can factor out an $\displaystyle s$ from the root on the right side of the equation:

$7\sqrt{3}=s\sqrt{3}$ $\displaystyle 7\sqrt{3}=s\sqrt{3}$

Just by looking at this, you can tell that the answer is:

s=7 $\displaystyle s=7$

Now, use the equation for the volume of a cube:

V = s^3 $\displaystyle V = s^3$

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

For our data, it is:

$\displaystyle V = 7^3 = 343$

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ISEE Upper Level Quantitative : Cubes

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 : Cubes

Example Question #1 : Cubes

Example Question #13 : Solid Geometry

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

Example Question #5 : Cubes

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

Example Question #1 : Cubes

Example Question #5 : How To Find The Surface Area Of A Cube

Example Question #6 : How To Find The Surface Area Of A Cube

All ISEE Upper Level Quantitative Resources

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