# ISEE Upper Level Quantitative : Cubes

## Example Questions

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

Which is the greater quantity?

(a) The volume of a cube with surface area  inches

(b) The volume of a cube with diagonal  inches

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(b) is greater.

Explanation:

The cube with the greater sidelength has the greater volume, so we need only calculate and compare sidelengths.

(a) , so the sidelength of the first cube can be found as follows:

inches

(b)  by an extension of the Pythagorean Theorem, so the sidelength of the second cube can be found as follows:

Since . The second cube has the greater sidelength and, subsequently, the greater volume. This makes (b) greater.

### Example Question #2 : 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

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It cannot be determined from the information given.

(a) is greater.

Explanation:

The sidelengths of Cubes 1, 2, 3, and 4 can be given values , respectively.

Then the volumes of the cubes are as follows:

Cube 1:

Cube 2:

Cube 3:

Cube 4:

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 .

(b) The sum of the volumes of Cubes 2 and 3 is .

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

### Example Question #3 : Cubes

What is the volume of a cube with side length  ? Round your answer to the nearest hundredth.

Explanation:

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

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

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

### Example Question #4 : Cubes

What is the volume of a cube on which one face has a diagonal of  ?

Explanation:

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

Notice that this makes a  triangle.

This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both , and the hypotenuse is .  This will allow us to make the proportion:

Multiplying both sides by , you get:

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

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

Now, rationalize the denominator:

### Example Question #5 : Cubes

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

Explanation:

The volume of a cube is defined by the formula

where  is the length of one side.

If , then

and

So one side measures 7 inches.

The surface area of a cube is defined by the formula

, so

The surface area is 294 square inches.

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

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

Explanation:

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

, where  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 () by  because a cube has  equal sides.

For our data, we know that ; therefore, our equation is:

### Example Question #7 : Cubes

What is the surface area of a cube with a volume  ?

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:

, where  is the side length.

For our data, this gives us:

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

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

, where  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 () by  because a cube has  equal sides.

For our data, this gives us:

### Example Question #8 : Cubes

What is the surface area of a cube with a volume  ?

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:

, where  is the side length.

For our data, this gives us:

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

(You will need to use a calculator for this.  If your calculator gives you something like  . . . 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:

, where  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 () by  because a cube has  equal sides.

For our data, this gives us:

### Example Question #2 : How To Find The Surface Area Of A Cube

What is the surface area for a cube with a diagonal length of  ?

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:

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

So, we know this, then:

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

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

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

, where  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 () by  because a cube has  equal sides.

For our data, this is:

### Example Question #21 : Solid Geometry

What is the volume of a cube with a diagonal length of  ?

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:

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

So, we know this, then:

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

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

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

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

For our data, it is:

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