Hi,

Can anyone please help me with the below question:

A 10-by-6 inch piece of paper is used to form the lateral surface of a cylinder. If the entire piece of paper is used to make the lateral surface, which of the following must be true of the two possible cylinders that can be formed?

A. The volume of the cylinder with height 10 is 60/Pi cubic inches greater than the volume of the cylinder with height 6.

B. The volume of the cylinder with height 6 is 60/Pi cubic inches greater than the volume of the cylinder with height 10.

C. The volume of the cylinder with height 10 is 60Pi cubic inches greater than the volume of the cylinder with height 6.

D. The volume of the cylinder with height 6 is 60Pi cubic inches greater than the volume of the cylinder with height 10.

E. The volume of the cylinder with height 6 is 240/Pi cubic inches greater than the volume of the cylinder with height 10.

The correct ans is B

The question stem mentions a 10 by 6 inch rectangular piece of paper used to construct the side of the cylinder. Keep in mind that there are two ways we can do this. With the 10 inch side as the height, imagine folding the piece of paper in a circular manner so that the two ends meet. The other way is to fold the paper with the 6 inch side as the height until the ends meet. Then the question becomes, what is the difference in the volume between the two?

Volume of a cylinder = Pi r^2 h. Here r is the radius and h is the height. Since we do not have the radius, we will need to calculate that somehow.

if you think about it, the length is the same thing as the circumference. You can see this visually when you fold the paper in the above described manner. Circumference is basically 2 Pi r. From this you can calculate r to be Circumference/(2 Pi)

Case 1: height = 10 and length (circumference) = 6 .
r = 6/(2 Pi) or 3/Pi
Vol = Pi (3/Pi)^2 * 10 = 90/Pi

Case 2: height = 6 and length (circumference) = 10
r = 10/(2 Pi) or 5/Pi
Vol = Pi (5/Pi)^2 * 6 = 150/Pi

Difference between the two (150/Pi) - (90/Pi) = 60/Pi
Answer b.


so r =

This explanation is good. Hope that makes sense.

can someone please explain why the radius is divided by pi? my answer is 60pi...

We have pi in the denominator because we divide the circumference by 2pi to determine the radius. Pi squared is in the denominator after we square the radius as part of the V formula.

When you divide 60 pi by pi squared, you'll get 60/pi.