# ACT Math : Spheres

## Example Questions

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

The surface area of a sphere is  feet. What is the radius?

Explanation:

Solve the equaiton for the surface area of a sphere for the radius and plug in the values:

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

What is the radius of a sphere with a volume of  ?  Round to the nearest hundredth.

Explanation:

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

For our data, we know:

Solve for . First, multiply both sides by :

Now, divide out the :

Using your calculator, you can solve for . Remember, if need be, you can raise  to the power of  if your calculator does not have a variable-root button.

This gives you:

If you get something like , just round up. This is a rounding issue with some calculators.

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

The volume of a sphere is . What is the diameter of the sphere? Round to the nearest hundredth.

Explanation:

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

For our data, we know:

Solve for . Begin by dividing out the  from both sides:

Next, multiply both sides by :

Using your calculator, solve for . Recall that you can always use the  power if you don't have a variable-root button.

You should get:

If you get , just round up to . This is a general rounding problem with calculators. Since you are looking for the diameter, you must double this to .

### Example Question #4 : How To Find The Radius Of A Sphere

What is the radius of a sphere with a surface area of  ?  Round to the nearest hundredth.

Explanation:

Recall that the surface area of a sphere is found by the equation:

For our data, this means:

Solve for . First, divide by :

Take the square root of both sides:

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

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

Explanation:

Given the volume of the sphere, , you need to use the formula for volume of a sphere  and work backwards to find the radius. I would multiply both sides by  to get rid of the  in the formula. You then have . Next, divide both sides by  so that all vyou have left is . Finally take the cube root of , to get  units for the radius.

### Example Question #6 : How To Find The Radius Of A Sphere

A cube with sides of  is circumscribed by a sphere, such that all eight vertices of the cube are tangent to the sphere. What is the sphere's radius?

Explanation:

Solving this problem requires recognizing that since the cube is circumscribed by the sphere, both solids share the same center. Now it is just a matter of finding the diagonal of the cube, which will double as the diameter of the sphere (by definition, any straight line which passes through the center of the sphere). The formula for the diagonal of a cube is , where  is the length of the side of a cube. (This occurs because you must use the Pythagorean theorem once for each 2-dimensional "corner" you travel to find the diagonal for a 3-dimensional shape, but for the ACT it's much faster to memorize the formula.)

In this case:

Since the radius is half the diameter, divide the result in half:

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

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

Explanation:

1. Use the volume to find the radius:

2. Use the radius to find the diameter:

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

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

Cannot be determined from the information given

Explanation:

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

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

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

Explanation:

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:

### Example Question #4 : How To Find The Diameter Of A Sphere

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.

Explanation:

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

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