All ACT Math Resources
Example Question #1 : Spheres
If a sphere has a volume of , what is its diameter?
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?
Cannot be determined from the information given
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 :
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.
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?
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.
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 .
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,