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

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Question

If a cube is inches tall, what is its volume?

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Answer

To find the volume of a cube, we multiply length by width by height, which can be represented with the forumla $v=l\cdot w\cdot h$ . Since a cube has equal sides, we can use for all three values.

$8\cdot 8\cdot 8=512: in ^{3}$

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Question

What is the volume of a cube with a side length equal to inches?

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Answer

The volume of a a cube (or rectangular prism) can be solved using the following equation:

$V=l\times\ w\times\ h$

$V=7in\times 7in\times $7in=343in^3$$

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Question

What is the volume of a cube in which the edge is equal to $\sqrt[3]{x}$ , and the value of is:

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Answer

First, the value of x must be solved for:

Given the edge of the cube is $\sqrt[3]{x}$ , plugging in the value of x results in $\sqrt[3]{62}$ . Thus, the area would be equal to this value cubed, which would result in 62.

Thus, 62 is the correct answer.

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Question

Give the volume of a cube with surface area 240 square inches.

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Answer

Let be the length of one edge of the cube. Since its surface area is 240 square inches, one face has one-sixth of this area, or $$\frac{240}{6}$ = 40$ square inches. Therefore, , and $s = $\sqrt{40}$ = 2$\sqrt{10}$$ .

The volume is the cube of this, or $V= \left ( 2$\sqrt{10}$ \right $)^{3}$= $2^{3}$ \cdot \left ( $\sqrt{10}$ \right $)^{3}$= 8 \cdot 10 $\sqrt{10}$= 80 $\sqrt{10}$$ cubic inches.

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Question

Give the volume of a cube with surface area 150 square inches.

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Answer

Let be the length of one edge of the cube. Since its surface area is 150 square inches, one face has one-sixth of this area, or $$\frac{150}{6}$ = 25$ square inches. Therefore, , and .

The volume is the cube of this, or $V= $5^{3}$=125$ cubic inches.

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Question

The distance from one vertex of a perfectly cubic aquarium to its opposite vertex is 1.5 meters. Give the volume of the aquarium in liters.

1 cubic meter = 1,000 liters.

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Answer

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

$s = $\sqrt{\frac{3}{4}$} = $\frac{\sqrt{3}$}{$\sqrt{4}$}} = $\frac{\sqrt{3}$}{2}$ meters.

Cube this sidelength to get the volume:

$s ^{3 } =\left ( $\frac{\sqrt{3}$}{2} \right $)^{3 }$= $\frac{\left(\sqrt{3}\right $)^{3 }$$}{2^{3 }$ } = $\frac{ 3\sqrt{3}$ }{8 }$ cubic meters.

To convert this to liters, multiply by 1,000:

$\frac{ 3\sqrt{3}$ }{8 } \cdot 1,000 = $\frac{ 3\sqrt{3}$ }{8 } \cdot $\frac{1,000}{1}$ = $\frac{ 3\sqrt{3}$ }{1 } \cdot $\frac{125}{1}$= 375$\sqrt{3}$ liters.

This is not among the given responses.

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Question

The length of a diagonal of one face of a cube is . Give the volume of the cube.

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Answer

Since a diagonal of a square face of the cube is, each side of each square has length $$\frac{N}{\sqrt{2}$}=\frac{N\sqrt{2}$}{2}$ .

Cube this to get the volume of the cube:

$\left($\frac{N\sqrt{2}$}{2} \right $)^{3}$$

$= $$\frac{N^{3}$$\left( $\sqrt{2}$ \right $)^{3}$ $}{2^{3}$ }$

$= $$\frac{N^{3}$$ \cdot 2 $\sqrt{2}$ }{8}$

$= $$\frac{N^{3}$$ $\sqrt{2}$ }{4}$

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Question

An aquarium is shaped like a perfect cube; the perimeter of each glass face is meters. If it is filled to the recommended capacity, then, to the nearest hundred cubic liters, how much water will it contain?

Note: $1\textup{ cubic meter}= 1\textup{,}000\textup{ cubic liters}$

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Answer

A perfect cube has square faces; if a face has perimeter meters, then each side of each face measures one fourth of this, or meters. The volume of the tank is the cube of this, or

cubic meters.

Its capacity in liters is $9.261 \times 1,000 = 9,261$ liters.

of this is

$9,261 \times 0.9 = 8,335$ liters.

This rounds to $8\textup,300$ liters, the correct response.

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Question

Your friend gives you a puzzle cube for your birthday. If the length of one edge is 5cm, what is the volume of the cube?

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Answer

Your friend gives you a puzzle cube for your birthday. If the length of one edge is 5cm, what is the volume of the cube?

To find the volume of a cube, use the following formula:

$V_${cube}=s^3$$

Where s is the side length.

Plug in what we know to get our answer:

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Question

A cube has a side length of , what is the volume of the cube?

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Answer

A cube has a side length of , what is the volume of the cube?

To find the volume of a cube, use the following formula:

$V_${cube}=s^3$$

Plug in our known side length and solve

$V_${cube}=(9mm)^3$$=729mm^3$$

Making our answer:

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Question

One of your holiday gifts is wrapped in a cube-shaped box.

If one of the edges has a length of 6 inches, what is the volume of the box?

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Answer

One of your holiday gifts is wrapped in a cube-shaped box.

If one of the edges has a length of 6 inches, what is the volume of the box?

Find the volume of a cube via the following:

$V_${cube}=s^3$$=(6in)^3$$=216in^3$$

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Question

Find the volume of a cube with a height of 3in.

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Answer

To find the volume of a cube, we will use the following formula:

where a is the length of any side of the cube.

Now, we know the height of the cube is 3in. Because it is a cube, all sides (lengths, widths, height) are the same. That is why we can find any length for the formula.

Knowing this, we can substitute into the formula. We get

$V = $(3$\text{in}$)^3$$

$V = 3$\text{in}$ \cdot 3$\text{in}$ \cdot 3$\text{in}$$

$V = $9$\text{in}$^2$ \cdot 3$\text{in}$$

$V = $27$\text{in}$^3$$

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Question

Find the area of a cube with a length of 5cm.

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Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the length of the cube is 5cm. Because it is a cube, all sides are equal. Therefore, the width and the height of the cube are also 5cm.

Knowing this, we will substitute into the formula. We get

$V = 5$\text{cm}$ \cdot5$\text{cm}$ \cdot5$\text{cm}$$

$V = $25$\text{cm}$^2$ \cdot 5$\text{cm}$$

$V = $125$\text{cm}$^3$$

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Question

Find the volume of a cube with a height of 8in.

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Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the height of the cube is 8in. Because it is a cube, all sides/lengths are equal. Therefore, the length and width are also 8in.

Knowing this, we can substitute into the formula. We get

$V = 8$\text{in}$ \cdot 8$\text{in}$ \cdot 8$\text{in}$$

$V = $64$\text{in}$^2$ \cdot 8$\text{in}$$

$V = $512$\text{in}$^3$$

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Question

While exploring an ancient ruin, you discover a small puzzle cube. You measure the side length to be . Find the cube's volume.

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Answer

While exploring an ancient ruin, you discover a small puzzle cube. You measure the side length to be . Find the cube's volume.

To find the volume of a cube, use the following formula:

So our answer is

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Question

You are building a box to hold your collection of rare rocks. You want to build a cube-shaped box with a side length of 3 feet. If you do so, what will the volume of your box be?

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Answer

You are building a box to hold your collection of rare rocks. You want to build a cube-shaped box with a side length of 3 feet. If you do so, what will the volume of your box be?

Begin with the formula for volume of a cube:

Where s is our side length and V is our volume.

Now, we need to plug in our side length and solve for V

So, our volume is

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Question

Find the volume of a cube with a width of 9in.

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Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the width is 9in. Because it is a cube, all sides are equal. Therefore, the length and the height are also 9in.

Knowing this, we can substitute into the formula. We get

$V = 9$\text{in}$ \cdot9$\text{in}$ \cdot9$\text{in}$$

$V = $81$\text{in}$^2$ \cdot 9$\text{in}$$

$V = $729$\text{in}$^3$$

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Question

Find the volume of a cube with a height of 8in.

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Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the height of the cube is 8in. Because it is a cube all sides (length, height, width) are the same. So, all are 8in.

Now we can substitute. We get

$V = 8$\text{in}$ \cdot 8$\text{in}$ \cdot 8$\text{in}$$

$V = $512$\text{in}$^3$$

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Question

Find the volume of a cube with a height of 7in.

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Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the height of the cube is 7in. Because it is a cube, all sides are equal. Therefore, all sides (the length, width, height) are all 7in.

So, we get

$V = 7$\text{in}$ \cdot 7$\text{in}$ \cdot 7$\text{in}$$

$V = $49$\text{in}$^2$ \cdot 7$\text{in}$$

$V = $343$\text{in}$^3$$

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Question

Find the volume of a cube with a length of 5in.

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Answer

To find the volume of a cube, we will use the following formula:

$V = l \cdot w \cdot h$

where l is the length, w is the width, and h is the height of the cube.

Now, we know the length of the cube is 5in. Because it is a cube, all sides/lengths are equal. Therefore, the length and width are also 5in.

Knowing this, we can substitute into the formula. We get

$V = 5$\text{in}$ \cdot 5$\text{in}$ \cdot 5$\text{in}$$

$V = $25$\text{in}$^2$ \cdot 5$\text{in}$$

$V = $125$\text{in}$^3$$

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