### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Cubes

Which is the greater quantity?

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

(b) The volume of a cube with diagonal inches

**Possible Answers:**

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

**Correct answer:**

(b) is greater.

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

**Possible Answers:**

It cannot be determined from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

**Correct answer:**

(a) is greater.

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.

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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:

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