### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #1 : Spheres

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 #1 : 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

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

**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 #2 : How To Find The Volume Of A Sphere

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.

(b) is greater.

It is impossible to tell from the information given.

(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.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

**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 #4 : How To Find The Volume Of A Sphere

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:**

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

(B) is greater

(A) 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 #3 : Spheres

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

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

(a) is the greater quantity

(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.

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