### All ACT Math Resources

## Example Questions

### Example Question #1 : Spheres

If a sphere has a volume of , what is its diameter?

**Possible Answers:**

**Correct answer:**

1. Use the volume to find the radius:

2. Use the radius to find the diameter:

### Example Question #2 : Spheres

A sphere has a volume of . What is its diameter?

**Possible Answers:**

Cannot be determined from the information given

**Correct answer:**

This question relies on knowledge of the formula for volume of a sphere, which is as follows:

In this equation, we have two variables, and . Additionally, we know that and is unknown. You can begin by rearranging the volume equation so it is solved for , then plug in and solve for :

Rearranged form:

Plug in for V

Simplify the part under the cubed root

1) Cancel the 's since they are in the numerator and denominator.

2) Simplify the fraction and the :

Thus we are left with

Then, either use your calculator and enter Or recall that in order to find that .

We're almost there, but we need to go a step further. Dodge the trap answer "" and carry on. Read the question carefully to see that we need the diameter, not the radius.

So

is our final answer.

### Example Question #3 : Spheres

A spherical plastic ball has a diameter of . What is the volume of the ball to the nearest cubic inch?

**Possible Answers:**

**Correct answer:**

To answer this question, we must calculate the volume of the ball using the equation for the volume of a sphere. The equation for the volume of a sphere is four-thirds multiplied by pi, which is then multiplied by the radius cubed. The equation can be written like this:

We are given the diameter of the sphere in the problem, which is . To get the radius from the diameter, we divide the diameter by . So, for this data:

We can then plug our newly found radius of two into the equation to find the volume. For this data:

We then multiply by .

We finally substitute 3.14 for pi and multiply again to get our answer.

The question asked us to round to the nearest whole cubic inch. To do this, we round a number up one place if the last digit is a 5, 6, 7, 8, or 9, and we round it down if the last digit is a 1, 2, 3, or 4. Therefore:

Therefore our answer is .

### Example Question #96 : Solid Geometry

A boulder breaks free on a slope and rolls downhill. It rolls for complete revolutions before grinding to a halt. If the boulder has a volume of cubic feet, how far in feet did the boulder roll? (Assume the boulder doesn't lose mass to friction). Round to 3 significant digits. Round your final answer to the nearest integer.

**Possible Answers:**

**Correct answer:**

The formula for the volume of a sphere is:

To figure out how far the sphere rolled, we need to know the circumference, so we must first figure out radius. Solve the formula for volume in terms of radius:

Since the answer asks us to round to the nearest integer, we are safe to round to at this point.

To find circumference, we now apply our circumference formula:

If our boulder rolled times, it covered that many times its own circumference.

Thus, our boulder rolled for

### Example Question #11 : Spheres

Find the diameter of a sphere whose radius is .

**Possible Answers:**

**Correct answer:**

To solve, simply remember that diameter is twice the radius. Don't be fooled when the radius is an algebraic expression and incorporates the arbitrary constant . Thus,

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