Rectangular Solids & Cylinders - 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 a rectangular prism has a length of , a width of , and a height of , what is the length of its diagonal?

Answer

The diagonal of a rectangular prism can be thought of as the hypotenuse of a right triangle formed by the height of the prism and the diagonal of its bottom face. First apply the Pythagorean Theorem to find the length of the diagonal of the bottom face, and then apply the Pythagorean Theorem again with this side and the height of the prism to find the length of its diagonal:

$C^2=52\rightarrow C=\sqrt{52}$

$3^2+(\sqrt{52})^2=D^2$

$D^2=61\rightarrow D=\sqrt{61}$

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Question

Calculate the diagonal length for a rectangular prism with a length of , a width of , and a height of .

Answer

The diagonal of a rectangular prism can be thought of as the hypotenuse of a right triangle, where the other two sides are the height of the prism and the diagonal of either its top or bottom face. This means we can find the length of the prism's diagonal using the Pythagorean Theorem, but first we must apply this theorem to find the diagonal of the prism's top or bottom face, which forms the base of the right triangle whose hypotenuse is the diagonal of the entire prism. After finding the face diagonal, we apply the Pythagorean Theorem again to calculate the answer:

$c^2=89\rightarrow c=\sqrt{89}$

$3^2+(\sqrt{89})^2=d^2$

$d^2=98\rightarrow d=\sqrt{98}=\sqrt{2\cdot 49}=7\sqrt{2}$

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Question

Calculate the diagonal length for a rectangular prism with a length of , a width of , and a height of .

Answer

The diagonal of a rectangular prism can be thought of as the hypotenuse of a right triangle whose other two sides are the height of the prism and the diagonal of either its top or bottom face. We start by finding the diagonal of either the prism's top or bottom face, as this is the base of the right triangle for which the diagonal of the prism is the hypotenuse:

$c^2=25\rightarrow c=5$

Now we apply the Pythagorean Theorem again, this time using the prism height and the face diagonal calculated above, and the hypotenuse we're left with is the same as the diagonal length of the rectangular prism:

$c^2=29\rightarrow c=\sqrt{29}$

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Question

A rectangular prism has a height of , a length of , and a width . What is the length of the prism's diagonal?

Answer

The diagonal of a rectangular prism is the hypotenuse of the right triangle formed by the height of the prism and the diagonal of its bottom face. Thus, we apply the Pythagorean Theorem twice: first to find the bottom face's diagonal, and again to find the diagonal of the prism. For the bottom face's diagonal , use the Pythagorean Theorem with the given length and width:

$l^{2} +w^{2}=c^{2}$

$\sqrt{l^{2} +w^{2}}=c^{2}$

$\sqrt{12^{2} +4^{2}}=c$

$\sqrt{144 +16}=c$

$\sqrt{160}=c$

$\sqrt{16\times 10}=c$

$4\sqrt{10}=c$

Using this value , we can now find the value of the prism's diagonal :

$h^{2}+c^{2}=d^{2}$

$\sqrt{h^{2} +c^{2}}=d$

$\sqrt{9^{2} +(\sqrt{160})^{2}}=d$

$\sqrt{81 +160}=d$

$\sqrt{241}=d$

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Question

A rectangular prism has a height of , a length of , and a width . What is the length of the prism's diagonal?

Answer

The diagonal of a rectangular prism is the hypotenuse of the right triangle formed by the height of the prism and the diagonal of its bottom face. Thus, we apply the Pythagorean Theorem twice: first to find the bottom face's diagonal, and again to find the diagonal of the prism. For the bottom face's diagonal , use the Pythagorean Theorem with the given length and width:

$l^{2} +w^{2}=c^{2}$

$\sqrt{l^{2} +w^{2}}=c^{2}$

$\sqrt{8^{2} +10^{2}}=c$

$\sqrt{64 +100}=c$

$\sqrt{164}=c$

$\sqrt{41\times 4}=c$

$2\sqrt{41}=c$

Using this value , we can now find the value of the prism's diagonal :

$h^{2}+c^{2}=d^{2}$

$\sqrt{h^{2} +c^{2}}=d$

$\sqrt{6^{2} +(\sqrt{164})^{2}}=d$

$\sqrt{36 +164}=d$

$\sqrt{200}=d$

$\sqrt{100\times 2}=d$

$10\sqrt{2}=d$

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Question

A rectangular prism has a height of , a length of , and a width . What is the length of the prism's diagonal?

Answer

The diagonal of a rectangular prism is the hypotenuse of the right triangle formed by the height of the prism and the diagonal of its bottom face. Thus, we apply the Pythagorean Theorem twice: first to find the bottom face's diagonal, and again to find the diagonal of the prism. For the bottom face's diagonal , use the Pythagorean Theorem with the given length and width:

$l^{2} +w^{2}=c^{2}$

$\sqrt{l^{2} +w^{2}}=c^{2}$

$\sqrt{7^{2} +4^{2}}=c$

$\sqrt{49 +16}=c$

$\sqrt{65}=c$

Using this value , we can now find the value of the prism's diagonal :

$h^{2}+c^{2}=d^{2}$

$\sqrt{h^{2} +c^{2}}=d$

$\sqrt{11^{2} +\sqrt{65}^{2}}=d$

$\sqrt{121+65}=d$

$\sqrt{186}=d$

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Question

Find the diagonal of a rectangular prism whose base is $3\times4$ and has a base of .

Answer

To solve, simply solve for the base diagonal which will become the side of the other triangle, whose hypotenuse is the diagonal we are looking for.

$c1^2=3^2+4^2\Rightarrow c=5$

$diagonal=\sqrt{5^2+12^2}=13$

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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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