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radius based on circumference

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by kalifalk » Sun Feb 05, 2012 9:11 pm
Hello,

I had the question below on my Manhattan Review practice test and I'm having a hard time understanding a concept.

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?

Answer:

One of the cylinders has a height of 6 and a base circumference of 10; the other has a height of 10 and a base circumference of 6.

The cylinder with a height of 6 and a base circumference of 10 has a radius of (5/pi ). Its volume is equal to r2h, or (5/pi )2(6) or 150/ .... etc.


What I don't understand is how a circle with a circumference of 10 has a radius of 5/pi... It may be simple, but I'm having a hard time making the connection.

Thanks much GMAT geniuses! :)
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by [email protected] » Sun Feb 05, 2012 9:29 pm
Lateral surface area of cylinder is =2*pi*r*h.
So, if the 10-by-6 inch piece of paper is used to form the lateral surface of a cylinder, then either the height of the cylinder could be 6 or 10.
When height=6,
then 2*pi*r=10 i.e. r=5/pi
When height=10,
then 2*pi*r=6 i.e. r=3/pi

Hope this helps.
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by Anurag@Gurome » Sun Feb 05, 2012 9:46 pm
kalifalk wrote:Hello,

I had the question below on my Manhattan Review practice test and I'm having a hard time understanding a concept.

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?

Answer:

One of the cylinders has a height of 6 and a base circumference of 10; the other has a height of 10 and a base circumference of 6.

The cylinder with a height of 6 and a base circumference of 10 has a radius of (5/pi ). Its volume is equal to r2h, or (5/pi )2(6) or 150/ .... etc.


What I don't understand is how a circle with a circumference of 10 has a radius of 5/pi... It may be simple, but I'm having a hard time making the connection.

Thanks much GMAT geniuses! :)
Lateral surface area of cylinder = 2(pi)rh, where r = radius of the base and top and h = height of the cylinder.
Circumference of circle = 2(pi)r

(1) When the circumference = 10 and height = 6
2(pi)r = 10
r = 5/pi
Volume of cylinder = (pi)r²h = (pi) * 25/(pi)² * 6 = 150/pi cubic inches

(2) When the circumference = 6 and height = 10
2(pi)r = 6
r = 3/pi
Volume of cylinder = (pi)r²h = (pi) * 9/(pi)² * 10 = 90/pi cubic inches

Difference between the two volumes = 150/pi - 90/pi = 60/pi
Therefore, volume of the cylinder with height 6 is 60/pi cubic inches greater than the volume of the cylinder with height 10.

Hope that helps.
Anurag Mairal, Ph.D., MBA
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Gurome, Inc.
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