### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #341 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Sphere A has volume . Sphere B has surface area . Which is the greater quantity?

(a) The radius of Sphere A

(b) The radius of Sphere B

**Possible Answers:**

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

**Correct answer:**

(b) is greater

(a) Substitute in the formula for the volume of a sphere:

inches

(b) Substitute in the formula for the surface area of a sphere:

inches

(b) is greater.

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

is a positive number. Which is the greater quantity?

(A) The surface area of a sphere with radius

(B) The surface area of a cube with edges of length

**Possible Answers:**

(A) is greater

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

(A) and (B) are equal

(B) is greater

**Correct answer:**

(B) is greater

The surface area of a sphere is times the square of its radius, which here is ; the surface area of the sphere in (A) is .

The area of one face of a cube is the square of the length of an edge, which here is , so the area of one face of the cube in (B) is . The cube has six faces so the total surface area is .

, so , giving the sphere less surface area. (B) is greater.

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

The height of Cone B is three times that of Cone A. The radius of the base of Cone B is one-half the radius of the base of Cone A.

Which is the greater quantity?

(a) The volume of Cone A

(b) The volume of Cone B

**Possible Answers:**

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(a) is greater.

Let be the radius and height of Cone A, respectively. Then the radius and height of Cone B are and , respectively.

(a) The volume of Cone A is .

(b) The volume of Cone B is

.

Since , the cone in (a) has the greater volume.

### Example Question #1 : Volume Of A Cone

The volume of a cone whose height is three times the radius of its base is one cubic yard. Give its radius in inches.

**Possible Answers:**

**Correct answer:**

The volume of a cone with base radius and height is

The height is three times this, or . Therefore, the formula becomes

Set this volume equal to one and solve for :

This is the radius in yards; since the radius in inches is requested, multiply by 36.

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

The height of a given cylinder is one half the height of a given cone. The radii of their bases are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

**Possible Answers:**

(b) is the greater quantity

(a) is the greater quantity

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

(a) and (b) are equal

**Correct answer:**

(b) is the greater quantity

Call the radius of the base of the cone and the height of the cone. The cylinder will have bases of radius and height .

In the formula for the volume of a cylinder, set and :

In the formula for the volume of a cone, set and :

, so

,

meaning that the cylinder has the greater volume.

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

The radius of the base of a given cone is three times that of each base of a given cylinder. The heights of the cone and the cylinder are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

**Possible Answers:**

(a) is the greater quantity

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

(a) and (b) are equal

(b) is the greater quantity

**Correct answer:**

(a) is the greater quantity

If we let be the radius of each base of the cylinder, then is the radius of the base of the cone. We can let be their common height.

In the formula for the volume of a cylinder, set and :

In the formula for the volume of a cone, set and :

, so . The cone has the greater volume.

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

What is the volume of a cylinder with a radius of 6 meters and a height of 11 meters? Use 3.14 for .

Note: The formula for the volume of a cylinder is:

**Possible Answers:**

**Correct answer:**

To calculate the volume, you must plug into the formula given in the problem. When you plug in, it should look like this: . Multiply all of these out and you get . The units are cubed because volume is always cubed.

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

The volume of a cylinder whose height is twice the diameter of its base is one cubic yard. Give its radius in inches.

**Possible Answers:**

**Correct answer:**

The volume of a cylinder with base radius and height is

The diameter of this circle is ; its height is twice this, or . Therefore, the formula becomes

Set this volume equal to one and solve for :

This is the radius in yards; multiply by 36 to get the radius in inches.

### Example Question #341 : Geometry

What is the height of a cylinder with a volume of and a radius of ?

**Possible Answers:**

**Correct answer:**

Recall that the equation of for the volume of a cylinder is:

For our values this is:

Solve for :

### Example Question #341 : Geometry

What is the volume of a cylinder with a height of in. and a radius of in?

**Possible Answers:**

**Correct answer:**

This is a rather direct question. Recall that the equation of for the volume of a cylinder is:

For our values this is:

This is the volume of the cylinder.

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