### All GRE Math Resources

## Example Questions

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

A cube with a surface area of 216 square units has a side length that is equal to the diameter of a certain sphere. What is the surface area of the sphere?

**Possible Answers:**

**Correct answer:**

Begin by solving for the length of one side of the cube. Use the formula for surface area to do this:

s= length of one side of the cube

The length of the side of the cube is equal to the diameter of the sphere. Therefore, the radius of the sphere is 3. Now use the formula for the surface area of a sphere:

The surface area of the sphere is .

### Example Question #1 : Spheres

The surface area of a sphere is . What is its diameter?

**Possible Answers:**

**Correct answer:**

The surface area of a sphere is defined by the equation:

For our data, this means:

Solving for , we get:

or

The diameter of the sphere is .

### Example Question #1 : Spheres

The volume of one sphere is . What is the diameter of a sphere of half that volume?

**Possible Answers:**

**Correct answer:**

Do not assume that the diameter will be half of the diameter of a sphere with volume of* *. Instead, begin with the sphere with a volume of* . *Such a simple action will prevent a vexing error!

Thus, we know from our equation for the volume of a sphere that:

Solving for , we get:

If you take the cube-root of both sides, you have:

First, you can factor out an :

Next, factor the :

Which simplifies to:

Thus, the diameter is double that or:

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

Find the surface area of a sphere with a diameter of 14. Use *π* = 22/7.

**Possible Answers:**

1256

872

428

616

2464

**Correct answer:**

616

Surface Area = 4*πr*^{2} = 4 * 22/7 * 7^{2} = 616

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

A sphere has a surface area of square inches. If the radius is doubled, what is the surface area of the larger sphere?

**Possible Answers:**

Cannot be determined

**Correct answer:**

The surface area of the larger sphere is NOT merely doubled from the smaller sphere, so we cannot double to find the answer.

We can use the surface area formula to find the radius of the original sphere.

*r*^{2} = 4

*r* = 2

Therefore the larger sphere has a radius of 2 * 2 = 4.

The new surface area is then square inches.

### Example Question #1 : Spheres

If a sphere has a volume of cubic inches, what is its surface area?

**Possible Answers:**

**Correct answer:**

The volume of a cube is equal to .

So we mutiply our volume by and divide by , giving us .

The surface area of a sphere is equal to , giving us .

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

How much does the volume of a sphere increase if its radius is increased by 50%?

**Possible Answers:**

0.3375%

50%

237.5%

150%

337.5%

**Correct answer:**

237.5%

Recall the equation for the volume of a sphere:

V = (4/3)πr^{3}

If we increase the radius by 50%, we can represent the new radius as being equal to r + 0.5r = 1.5r.

Replace this into the equation for the volume and simplify:

V_{2} = (4/3)π(1.5r)^{3} = (4/3)π(3.375r^{3})

Rewrite this so that you can compare the two volumes:

V_{2 }= 3.375 * (4/3)πr^{3} = 3.375 * [(4/3)πr^{3}]

This is the same as:

V_{2} = 3.375 * V

This means that the new volume is 337.5% of the original. However, note that the question asked for the increase, which would be an increase by 237.5%.

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

A cube weighs 216 grams. If you carve a sphere out of the cube such that the diameter of the sphere is equal to one of the sides of the square, how many grams is the weight of the resulting sphere?

**Possible Answers:**

9π

144π

216π

36π

288π

**Correct answer:**

36π

Remember that the weight of an object is analogous to the volume. Since the weight of the sphere is 216, the volume of the sphere is proportional to 216. Remember the equation for volume of a sphere:

V = a * a * a = 216

Take the cube root of 216 to find that the length of one of the cubes is proportional to 6. According to the question, one of the sides of the cube is equivalent to the diameter of the sphere.

Thus d = 6 and r = d/2 = 3 for the sphere.

Remember the volume equation for a sphere:

V = 4/3 * π * r^{3}

Plug in r = 3 to find V = 36π

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

What is the volume of a sphere with a radius of 3?

**Possible Answers:**

36*π*

24*π*

96*π*

112*π*

48*π*

**Correct answer:**

36*π*

Volume of a sphere = 4/3 * *πr*^{3} = 4/3 * *π* * 3^{3 }= 36*π*

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

How many times greater is the volume of a sphere with radius of 3 than the volume of a sphere with radius of ?

**Possible Answers:**

**Correct answer:**

The formula for the volume of a sphere is .

Use this to find the volume of each sphere, then take the quotient of the two volumes to determine the relationship between the two.

For the first sphere, .

For the second sphere, .

Since both volumes have factors of 4/3 and pi, we'll ignore those and divide by the remaining factors. Keep in mind that the volume of the first sphere must be in the numerator due to the wording of the question.