### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #321 : Geometry

What is the length of one side of a cube that has a surface area of ?

**Possible Answers:**

**Correct answer:**

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.

Now, we know that is ; therefore, we can write:

Solve for :

Take the square root of both sides:

This is the length of one of your sides.

### Example Question #1 : Tetrahedrons

Which is the greater quantity?

(a) The surface area of a regular tetrahedron with edges of length 1

(b) 2

**Possible Answers:**

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

**Correct answer:**

(b) is greater.

A regular tetrahedron has four faces, each of which is an equilateral triangle. Therefore, its surface area, given sidelength , is

.

Substitute :

, so (b) is greater.

### Example Question #1 : Spheres

The volume of a sphere is one cubic yard. Give its radius in inches.

**Possible Answers:**

**Correct answer:**

The volume of a sphere with radius is

.

To find the radius in yards, we set and solve for .

yards.

Since the problem requests the radius in inches, multiply by 36:

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

In terms of , give the volume, in cubic feet, of a spherical tank with diameter 36 inches.

**Possible Answers:**

**Correct answer:**

36 inches = feet, the diameter of the tank. Half of this, or feet, is the radius. Set , substitute in the volume formula, and solve for :

### Example Question #2 : How To Find The Volume Of A Sphere

Which is the greater quantity?

(a) The volume of a sphere with radius

(b) The volume of a cube with sidelength

**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

A sphere with radius has diameter and can be inscribed inside a cube of sidelength . Therefore, the cube in (b) has the greater volume.

### Example Question #31 : Solid Geometry

Which is the greater quantity?

(a) The volume of a cube with sidelength inches.

(b) The volume of a sphere with radius inches.

**Possible Answers:**

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

**Correct answer:**

(a) is greater.

You do not need to calculate the volumes of the figures. All you need to do is observe that a sphere with radius inches has diameter inches, and can therefore be inscribed inside the cube with sidelength inches. This give the cube larger volume, making (a) the greater quantity.

### Example Question #3 : How To Find The Volume Of A Sphere

Which is the greater quantity?

(a) The volume of a sphere with diameter one foot

(b)

**Possible Answers:**

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(a) is greater.

The radius of the sphere is one half of its diameter of one foot, which is six inches, so substitute :

cubic inches,

which is greater than .

### Example Question #1 : Spheres

is a positive number. Which is the greater quantity?

(A) The volume of a cube with edges of length

(B) The volume of a sphere with radius

**Possible Answers:**

(A) is greater

It is impossible to determine which is greater from the information given

(B) is greater

(A) and (B) are equal

**Correct answer:**

(A) is greater

No calculation is really needed here, as a sphere with radius - and, subsequently, diameter - can be inscribed inside a cube of sidelength . This makes (A), the volume of the cube, the greater.

### Example Question #5 : How To Find The Volume Of A Sphere

Which is the greater quantity?

(a) The radius of a sphere with surface area

(b) The radius of a sphere with volume

**Possible Answers:**

(a) and (b) are equal

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

**Correct answer:**

(a) and (b) are equal

The formula for the surface area of a sphere, given its radius , is

The sphere in (a) has surface area , so

The formula for the volume of a sphere, given its radius , is

The sphere in (b) has volume , so

The radius of both spheres is 3.

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

In terms of , give the surface area, in square inches, of a spherical water tank with a diameter of 20 feet.

**Possible Answers:**

**Correct answer:**

feet = inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set , substitute in the surface area formula, and solve for :