Cubes - ISEE Upper Level Quantitative Reasoning

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Question

The volume of a cube is . What is the length of an edge of the cube?

Answer

Let be the length of an edge of the cube. The volume of a cube can be determined by the equation:

$\sqrt[3]{x^3}=\sqrt[3]{64}$

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Question

The above cube has surface area 486. Evaluate .

Answer

The surface area of a cube is six times the square of the length of each edge, which here is . Therefore,

$6s^{2} = A$

Substituting, then solving for :

$6 (x-7)^{2} = 486$

$6 (x-7)^{2} \div 6 = 486 \div 6$

$(x-7)^{2} = 81$

$\sqrt{(x-7)^{2} }= \sqrt{81}$

Since the sidelength is positive,

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Question

There is a sculpture in front of town hall which is shaped like a cube. If it has a volume of , what is the length of one side of the cube?

Answer

There is a sculpture in front of town hall which is shaped like a cube. If it has a volume of , what is the length of one side of the cube?

To find the side length of a cube from its volume, simply use the following formula:

$V_{cube}=s^3$

Plug in what is known and use some algebra to get our answer:

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

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Question

You have a cube with a volume of . What is the cube's side length?

Answer

You have a cube with a volume of . What is the cube's side length?

If we begin with the formula for volume of a cube, we can work backwards to find the side length.

$V_{cube}=s^3$

$s=\sqrt[3]{125m^3}=5m$

Making our answer:

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Question

You have a crate with equal dimensions (height, length and width). If the volume of the cube is , what is the length of one of the crate's dimension?

Answer

You have a crate with equal dimensions (height, length and width). If the volume of the cube is , what is the length of one of the crate's dimension?

Let's begin with realizing that we are dealing with a cube. A crate with equal dimensions will have equal height, length, and width, so it must be a cube.

With that in mind, we can find our side length by starting with the volume and working backward.

So, to find our side length, we just need to take the cubed root of the volume.

$s=\sqrt[3]{216m^3}=6m$

So, our answer is 6 meters, a fairly large crate!

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Question

If the volume of a cube is 30, what must be the length of the edge of the cube?

Answer

Write the formula to find the volume of a cube.

Substitute the volume into the equation.

Cube root both sides.

$\sqrt[3]{30}=\sqrt[3]{s^3}$

The answer is: $\sqrt[3]{30}$

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Question

You have a cube with a volume of . What is the cube's side length?

Answer

You have a cube with a volume of . What is the cube's side length?

If we begin with the formula for volume of a cube, we can work backwards to find the side length.

$V_{cube}=s^3$

$s=\sqrt[3]{125m^3}=5m$

Making our answer:

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Question

The length of the side of a cube is . Give its surface area in terms of .

Answer

Substitute in the formula for the surface area of a cube:

$A = 6s^{2}$

$A = 6 \left ( x - 5\right )^{2}$

$A = 6 \left ( x^{2} - 2 \cdot 5 \cdot x + 5^{2}\right )$

$A = 6 \left ( x^{2} - 10x +25\right )$

$A = 6 x^{2} - 60x +150$

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Question

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

Answer

To find the surface area of a cube, use the formula $6s^{2}$ , where represents the length of the side. Since the side of the cube measures , we can substitute in for .

$6(4)^{2}=96: cm^{2}$

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Question

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

Answer

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

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

$SA_{cube}=6s^2$

This works, because we have 6 sides, each of which has an area of

Plug in our known to get our answer:

$SA_{cube}=6(5cm)^2=6*25cm^2=150cm^2$

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Question

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

Answer

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

Surface area of a cube can be found as follows:

$SA_{cube}=6s^2$

Plug in our side length to find our answer:

$SA_{cube}=6(9mm)^2=6*81mm^2=486mm^2$

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 surface area of the box?

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 surface area of the box?

We can find the surface area of a square by squaring the length of the side and then multiplying it by 6.

$SA_{square}=6s^2=6(6in)^2=216in^2$

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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 surface area.

Answer

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

To find the surface area, use the following formula:

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Question

Find the surface area of a cube that has a width of 6cm.

Answer

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

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

Now, we know the width of the cube is 6cm. Because it is a cube, all sides are 6cm. That is why we can choose any side to substitute into the formula.

Now, knowing this, we can substitute into the formula. We get

$SA = 6 \cdot (6\text{cm})^2$

$SA = 6 \cdot 36\text{cm}^2$

$SA = 216\text{cm}^2$

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Question

Find the surface area of a cube with a length of 7in.

Answer

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

where a is the length of any side of the cube. Note that all sides are equal on a cube. That is why we can use any length in the formula.

Now, we know the length of the cube is 7in.

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

$SA = 6 \cdot (7\text{in})^2$

$SA = 6 \cdot 49\text{in}^2$

$SA = 294\text{in}^2$

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Question

Find the surface area of a cube with a width of 4cm.

Answer

To find the surface area of a cube, we will use the following formula.

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

Now, we know the width of the cube is 4in. Because it is a cube, all sides are equal (this is why we can use any length in the formula). So, we will use 4in in the formula. We get

$SA = 6 \cdot (4\text{in})^2$

$SA = 6 \cdot 16\text{in}^2$

$SA = 96\text{in}^2$

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Question

Find the surface area of a cube with a length of 12in.

Answer

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

$SA = 6 \cdot l \cdot w$

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

Now, we know the length of the cube is 12in. Because it is a cube, all lengths, widths, and heights are the same. Therefore, the width is also 12in.

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

$SA = 6 \cdot 12\text{in} \cdot 12\text{in}$

$SA = 6 \cdot 144\text{in}^2$

$SA = 864\text{in}^2$

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Question

Find the surface area of a cube with a base of 8in.

Answer

A cube has all equal sides (length, width, height). To find the surface area of a cube, we will use the following formula:

Now, we know the base of the cube is 8in. Because all sides are equal (i.e. all sides are 8in), we will use that to substitute into the formula. We get

$SA = 6 \cdot (8\text{in})^2$

$SA = 6 \cdot 64\text{in}^2$

$SA = 384\text{in}^2$

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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 surface area of your box be?

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 surface area of your box be?

To find the surface area, use the formula:

Where s is our side length and SA is our surface area.

Plug in our known and simplify

So, our answer is

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Question

Find the surface area of a cube with a length of 11in.

Answer

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

where a is the length of any side of the cube. Because it is a cube, all sides are equal. Therefore, we can use any side in the formula.

Now, we know the length of the cube is 11in. We can substitute that into the formula. We get

$SA = 6 \cdot (11\text{in})^2$

$SA = 6 \cdot 121\text{in}^2$

$SA = 726\text{in}^2$

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