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A 10-by-6 inch piece of paper is used to form

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A 10-by-6 inch piece of paper is used to form [#permalink]

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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 \(\frac{60}{\pi}\) cubic inches greater than the volume of the cylinder with height 6.

B. The volume of the cylinder with height 6 is \(\frac{60}{\pi}\) cubic inches greater than the volume of the cylinder with height 10.

C. The volume of the cylinder with height 10 is \(60\pi\) cubic inches greater than the volume of the cylinder with height 6.

D. The volume of the cylinder with height 6 is \(60\pi\) cubic inches greater than the volume of the cylinder with height 10.

E. The volume of the cylinder with height 6 is \(\frac{240}{\pi}\) cubic inches greater than the volume of the cylinder with height 10.
[Reveal] Spoiler: OA

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Last edited by Bunuel on 21 Jun 2014, 07:43, edited 1 time in total.
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Re: A 10-by-6 inch piece of paper is used to form [#permalink]

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New post 10 Jan 2011, 20:11
Simple math
1. Answer choices asking about difference in volumes
2. Find volume (pi)*r^2*h of each
case a: 2(pi)r =10 and h = 6 --> (pi)r^2*h = (pi) [10/2*(pi)]^2 * 6 = 25*6/(pi) = 150/pi
case b: 2(pi)r =6 and h = 10 --> (pi)r^2*h = (pi) [6/2*(pi)]^2 * 10 = 9*10/(pi) = 90/pi
3. Difference is 60/pi
4. So [case a volume] when h = 6 is greater than [case b volume ] when h = 10, by 60/(pi)
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Re: A 10-by-6 inch piece of paper is used to form [#permalink]

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New post 21 Jun 2014, 03:32
Bunuel , can you please explain this problem. I am not able to understand the difference between Option B and D. According to me option D should be correct but its not.
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Re: A 10-by-6 inch piece of paper is used to form [#permalink]

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gauravsoni wrote:
Bunuel , can you please explain this problem. I am not able to understand the difference between Option B and D. According to me option D should be correct but its not.


The difference is that B says "\(\frac{60}{\pi}\)", while D says: "\(60\pi\)". Formatted the original post to make it clearer.

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 \(\frac{60}{\pi}\) cubic inches greater than the volume of the cylinder with height 6.

B. The volume of the cylinder with height 6 is \(\frac{60}{\pi}\) cubic inches greater than the volume of the cylinder with height 10.

C. The volume of the cylinder with height 10 is \(60\pi\) cubic inches greater than the volume of the cylinder with height 6.

D. The volume of the cylinder with height 6 is \(60\pi\) cubic inches greater than the volume of the cylinder with height 10.

E. The volume of the cylinder with height 6 is \(\frac{240}{\pi}\) cubic inches greater than the volume of the cylinder with height 10.

We can make 2 cylinders:

With height of 6 and the radius of the base of \(r=\frac{5}{\pi}\) (from \(2\pi{r}=10\) --> \(r=\frac{5}{\pi}\)) --> \(volume=\pi{r^2}h=\frac{150}{\pi}\).

With height of 10 and the radius of the base of \(r=\frac{3}{\pi}\) (from \(2\pi{r}=6\) --> \(r=\frac{3}{\pi}\)) --> \(volume=\pi{r^2}h=\frac{90}{\pi}\).

The volume of the first one is \(\frac{60}{\pi}\) cubic inches greater than the volume of the second one.

Answer: B.


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Re: A 10-by-6 inch piece of paper is used to form [#permalink]

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New post 22 Jun 2014, 01:23
Bunuel wrote:
gauravsoni wrote:
Bunuel , can you please explain this problem. I am not able to understand the difference between Option B and D. According to me option D should be correct but its not.


The difference is that B says "\(\frac{60}{\pi}\)", while D says: "\(60\pi\)". Formatted the original post to make it clearer.

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 \(\frac{60}{\pi}\) cubic inches greater than the volume of the cylinder with height 6.

B. The volume of the cylinder with height 6 is \(\frac{60}{\pi}\) cubic inches greater than the volume of the cylinder with height 10.

C. The volume of the cylinder with height 10 is \(60\pi\) cubic inches greater than the volume of the cylinder with height 6.

D. The volume of the cylinder with height 6 is \(60\pi\) cubic inches greater than the volume of the cylinder with height 10.

E. The volume of the cylinder with height 6 is \(\frac{240}{\pi}\) cubic inches greater than the volume of the cylinder with height 10.

We can make 2 cylinders:

With height of 6 and the radius of the base of \(r=\frac{5}{\pi}\) (from \(2\pi{r}=10\) --> \(r=\frac{5}{\pi}\)) --> \(volume=\pi{r^2}h=\frac{150}{\pi}\).

With height of 10 and the radius of the base of \(r=\frac{3}{\pi}\) (from \(2\pi{r}=6\) --> \(r=\frac{3}{\pi}\)) --> \(volume=\pi{r^2}h=\frac{90}{\pi}\).

The volume of the first one is \(\frac{60}{\pi}\) cubic inches greater than the volume of the second one.

Answer: B.




Thanks Bunuel, I thought of using the width of the rectangle as the radius, now i see that its actually the circumference of the circular base. Thanks for the clarification.
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Re: A 10-by-6 inch piece of paper is used to form [#permalink]

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New post 21 Sep 2014, 00:03
HI Bunuel,

Can you pls explain how can i know its the circumference of the base? Thank you very much.
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Re: A 10-by-6 inch piece of paper is used to form [#permalink]

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vietnammba wrote:
HI Bunuel,

Can you pls explain how can i know its the circumference of the base? Thank you very much.


When you roll a paper into a cylinder one of the dimensions of the paper becomes the height of the cylinder and the another forms circumference of the base:
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New post 21 Sep 2014, 00:30
I get it now, thank you :)
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Re: A 10-by-6 inch piece of paper is used to form [#permalink]

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New post 08 Jan 2016, 19:31
we have 2 possible cylinders:
h=10, or h=6.
now, with radius, it is more tricky.
We know that Circumference is 2piR.
now, if h=10, then 2piR=6, or R=3/pi
if h=6, then 2piR=10, or 5/pi.

volume is pi*r^2*h.
1. h=10, r=3/pi. volume is 90/pi
2. h=6, r=5/pi. volume is 150/pi.

we can eliminate right away A, C, and E. - A, B because the cylinder with h=6 is greater. E because the difference is 60/pi and not 240/pi.
D is incorrect because it tells 60pi, when actually it is 60/pi.


B thus is the correct answer.
Re: A 10-by-6 inch piece of paper is used to form   [#permalink] 08 Jan 2016, 19:31
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