Cubes - GMAT Quantitative Reasoning

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Question

What is the length of the diagonal of a cube if its side length is ?

Answer

The diagonal of a cube extends from one of its corners diagonally through the cube to the opposite corner, so it can be thought of as the hypotenuse of a right triangle formed by the height of the cube and the diagonal of its base. First we must find the diagonal of the base, which will be the same as the diagonal of any face of the cube, by applying the Pythagorean theorem:

$c^2=18\rightarrow c=\sqrt{18}=3\sqrt{2}$

Now that we know the length of the diagonal of any face on the cube, we can use the Pythagorean theorem again with this length and the height of the cube, whose hypotenuse is the length of the diagonal for the cube:

$(3\sqrt{2})^2+3^2=d^2$

$d^2=27\rightarrow d=\sqrt{27}=3\sqrt{3}$

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Question

is a cube and face has an area of . What is the length of diagonal of the cube ?

Answer

To find the diagonal of a cube we can apply the formula $d=e\sqrt{3}$ , where is the length of the diagonal and where is the length of an edge of the cube.

Since we are given an area of a face of the cube, we can find the length of an edge simply by taking its square root.

$9=s^2 \rightarrow \sqrt 9 =s \rightarrow 3=s$

Here the length of an edge is 3.

Thefore the final andwer is $d=e\sqrt3=3\sqrt{3}$ .

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Question

What is the length of the diagonal of cube , knowing that face has diagonal equal to ?

Answer

To find the length of the diagonal of the cube, we can apply the formula, however, we firstly need to find the length of an edge, by applying the formula for the diagonal of the square.

$d=s\sqrt{2}$ where is the diagonal of face ABCD, and , the length of one of the side of this square.

The length of must be $\sqrt{2}$ , which is the length of the edges of the square.

Therefore we can now use the formula for the length of the diagonal of the cube:

$e\sqrt{3}$ , where is the length of an edge.

Since $e=\sqrt{2}$ , we get the final answer $\sqrt{6}$ .

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Question

If a cube has a side length of , what is the length of its diagonal?

Answer

The diagonal of a cube is the hypotenuse of a right triangle whose height is one side and whose base is the diagonal of one of the faces. First we must use the Pythagorean theorem to find the length of the diagonal of one of the faces, and then we use the theorem again with this value and length of one side of the cube to find the length of its diagonal:

$c^2=32\rightarrow c=\sqrt{32}$

So this is the length of the diagonal of one of the faces, which we plug into the Pythagorean theorem with the length of one side to find the length of the diagonal for the cube:

$4^2+(\sqrt{32})^2=d^2$

$d^2=48\rightarrow d=\sqrt{48}=\sqrt{3*16}=4\sqrt{3}$

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Question

A given cube has an edge length of . What is the length of the diagonal of the cube?

Answer

The diagonal of a cube with an edge length can be defined by the equation $d=s\sqrt{3}$ . Given in this instance, $d=5\sqrt{3}$ .

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Question

A given cube has an edge length of . What is the length of the diagonal of the cube?

Answer

The diagonal of a cube with an edge length can be defined by the equation $d=s\sqrt{3}$ . Given in this instance, $d=9\sqrt{3}$ .

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Question

A given cube has an edge length of $\sqrt{3}$ . What is the length of the diagonal of the cube?

Answer

The diagonal of a cube with an edge length can be defined by the equation $d=s\sqrt{3}$ . Given $s=\sqrt{3}$ in this instance, $d=\sqrt{3}\left ( \sqrt{3} \right )=3$ .

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Question

If the volume of a cube is units cubed, what is the length of each side of the cube?

Answer

We solve for the side of the cube by deriving it from the volume of a cube formula:

$s=\sqrt[3]{V}$

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

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Question

If a cube has a surface area of , what is the length of one side?

Answer

To find the surface area of a cube, use the following formula:

Where s is our side length.

Rearrange for s to get the following:

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

Plug in 54 for SA and solve

$s=\sqrt{\frac{54}{6}}=\sqrt{9}=3$

Don't forget your units, in this case centimeters so we get 3 centimeters

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Question

If a cube has a surface area of , what is the length of one of its sides?

Answer

To solve for the side length of the cube, we need to use the formula for surface area. There are six faces on a cube, so its total surface area is just six times the area of one of its square faces, which is given by the side length squared:

$L^2=\frac{216}{6}=36$

$L=\sqrt{36}=6$

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Question

If a cube has a volume of and is made up of smaller cubes, what is the length of one side of one of the smaller cubes?

Answer

Volume of a cube is equal to the cube of its side. 343 is equal to 7 cubed, so the length of the whole cube is 7 cm. The cube is made up of 27 smaller cubes though. That means that one face of the cube is made up of 9 cubes and one edge of the whole cube is made up of 3 small cubes. That means that the length of one small cube is equal to the total length of one side divided by the numbers of cubes per side

$\small V=s^3$

$\small \sqrt[3]{343}=7$

Then we get our final answer by doing:

$\small \frac{7}{3}cm$

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Question

is a cube with diagonal . What is the length of an edge of the cube?

Answer

We are given the length of the diagonal of the cube.

Therefore we can find the length of an edge by using the formula

$d=e\sqrt{3}$ or $e=\frac{d}{\sqrt{3}}$ , and $\frac{3}{\sqrt{3}}=\frac{3}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=\sqrt{3}$ .

Therefore, the final answer is $\sqrt{3}$ .

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Question

Find the length of the edge of a cube given that the volume is .

Answer

To find side length, you must use the equation for volume of a cube and solve for .

$V=s^3\Rightarrow s=\sqrt[3]{V}$

Thus,

$s=\sqrt[3]{64}=4$

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Question

What is the surface area of a box that is 3 feet long, 2 feet wide, and 4 feet high?

Answer

$\dpi{100} \small SA = 2lw + 2lh + 2wh = 2\times 3\times 2 + 2 \times 3\times 4 + 2\times 2\times 4 = 52$

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Question

What is the surface area of a cube with side length 4?

Answer

$SA=6\ast side^{2}=6\ast 4^{2}=6\ast 16=96$

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Question

The surface area of a certain cube is 150 square feet. If the width of the cube is increased by 2 feet, the length decreased by 2 feet and the height increased by 1 foot, what is the new surface area?

Answer

The first step to answering this qestion is to determine the original length of the sides of the cube. The surface area of a cube is given by:

$S_{A} = 6x^{2}$

Where is the length of each side. This tells us that for our cube:

$150 = 6x^{2}$ ==> $25 = x^{2}$ ==>

If the width increases by 2, the length decreases by 2 and the height increases by 1:

, ,

We now have a rectangular prism. The surface area of a rectangular prism is given by:

$S_{A} = 2(wl + lh + hw)$

For our prism:

$2((7\times 3) + (7 \times 6) + (6 \times 3)) = 2(21 + 42 + 18) = 162$

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Question

What is the surface area of a cube with a side length of ?

Answer

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Question

A cube is inscribed inside a sphere with surface area $100 \pi$ . Give the volume of the cube.

Answer

Each diagonal of the inscribed cube is a diameter of the sphere, so its length is the sphere's diameter, or twice its radius.

The sphere has surface area $80 \pi$ , so the radius is calculated as follows:

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

$4 \pi r^{2} = 100 \pi$

$r^{2} =100 \pi \div 4 \pi = 25$

The diameter of the circle - and the length of a diagonal of the cube - is twice this, or 10.

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

$s^{2}+s^{2}+s^{2}=10^{2}$

$3s^{2}=100$

$s^{2}=\frac{100}{3}$

$s = \sqrt{\frac{100}{3}} = \frac{\sqrt{100}}{\sqrt{3}} = \frac{10}{\sqrt{3}}$

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

$V=\left ( \frac{10}{\sqrt{3}} \right )^{3} = \frac{10^{3} } {\left (\sqrt{3} \right )^{3} } =\frac{1,000 } {3 \sqrt{3} } =\frac{1,000 \cdot \sqrt{3} } {3 \sqrt{3} \cdot \sqrt{3} } =\frac{1,000 \sqrt{3} } {9 }$

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Question

A sphere of volume $28 8 \pi$ is inscribed inside a cube. Give the surface area 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 volume formula:

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

$288 \pi= \frac{4}{3} \pi r^{3}$

$\frac{4}{3} r^{3} = 288$

$\frac{3}{4} \cdot \frac{4}{3} r^{3} = \frac{3}{4} \cdot 288$

$r^{3} = 216$

Twice this, or 12, is the diameter, and, subsequently, the length of an edge of the cube. If , the surface area is

$A = 6s^{2} = 6 \cdot 12^{2}= 6 \cdot 144 = 864$

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Question

A cube is inscribed inside a sphere of volume $2 88 \pi$ . Give the surface area 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 volume formula:

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

$288 \pi= \frac{4}{3} \pi r^{3}$

$\frac{4}{3} r^{3} = 288$

$\frac{3}{4} \cdot \frac{4}{3} r^{3} = \frac{3}{4} \cdot 288$

$r^{3} = 216$

Twice this, or 12, is the diameter, and, subsequently, the length of a diagonal of the cube. By an extension of the Pythagorean Theorem, if is the length of an edge of the cube,

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

$3s^{2} = 144$

$s^{2} = 48$

The surface area is six times this:

$SA = 6s^{2} = 6 \cdot 48 = 288$

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