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655-705 Level|   Geometry|      
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ajit257
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A 10-by-6 inch piece of paper is used to form the lateral surface of a Updated on: 02 Jan 2017, 04:55
Originally posted by ajit257 on 10 Jan 2011, 19:00.
Last edited by Bunuel on 02 Jan 2017, 04:55, edited 2 times in total.
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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.
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B
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Bunuel
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A 10-by-6 inch piece of paper is used to form the lateral surface of a 21 Sep 2014, 00:22
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vietnammba
HI Bunuel,

Can you pls explain how can i know its the circumference of the base? Thank you very much.
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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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BenchPrepGURU
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A 10-by-6 inch piece of paper is used to form the lateral surface of a 28 Jul 2011, 13:05
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Here's the 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.



I am having difficulty with this one: Please help and explain, if answer if B or E, WHY it is (#/pi) and not #pi, considering pi(r-squared)(h) is the volume formula

Thanks,
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Hopefully, you have some intuition about which of the possible cylinders is going to have the greater volume. Because the volume of a cylinder is directly proportional to the height and directly proportional to the square of the radius, the size of the radius has the greatest effect on the volume of the cylinder. Given this intuition we can eliminate A and C.

The answer given already is correct, but I'm going to provide some more details in term of calculation

Cylinder with height 10 and circumference 6:
\(2pi(r) = 6\)
\(r = \frac{3}{(pi)}\)
\(V = (pi)(\frac{3}{pi})^2(10)\)
\(V = \frac{90}{pi}\)

Cylinder with height 6 and circumference 10:
\(2pi(r) = 10\)
\(r = \frac{5}{(pi)}\)
\(V = (pi)(\frac{5}{pi})^2(6)\)
\(V = \frac{150}{pi}\)

150-90 = 60

So, B.
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A 10-by-6 inch piece of paper is used to form the lateral surface of a 10 Jan 2011, 20:11
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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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A 10-by-6 inch piece of paper is used to form the lateral surface of a 21 Jun 2014, 03:32
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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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A 10-by-6 inch piece of paper is used to form the lateral surface of a 21 Jun 2014, 07:59
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gauravsoni
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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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.

Theory on Geometry:
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A 10-by-6 inch piece of paper is used to form the lateral surface of a 22 Jun 2014, 01:23
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Bunuel
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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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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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A 10-by-6 inch piece of paper is used to form the lateral surface of a 21 Sep 2014, 00:03
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HI Bunuel,

Can you pls explain how can i know its the circumference of the base? Thank you very much.
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A 10-by-6 inch piece of paper is used to form the lateral surface of a Updated on: 11 Nov 2017, 08:09
Originally posted by OCDianaOC on 10 Nov 2017, 19:25.
Last edited by OCDianaOC on 11 Nov 2017, 08:09, edited 1 time in total.
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V = pi(5/pi)^2 x 6 = 150/pi. I'm confused as too why two pi turns into 150/pi.

V = pi(3/pi)^2 x 10 = 90/pi. I'm confused as too why two pi turns into 90/pi.

Can someone explain? Aren't we supposed to cross-cancel the pi out?
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A 10-by-6 inch piece of paper is used to form the lateral surface of a 11 Nov 2017, 01:37
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OCDianaOC
V = pi(5/pi)*2 x 6 = 150/pi. I'm confused as too why two pi turns into 150/pi.

V = pi(3/pi)*2 x 10 = 90/pi. I'm confused as too why two pi turns into 90/pi.

Can someone explain? Aren't we supposed to cross-cancel the pi out?
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The volume formula is \(volume=\pi{r^2}h\). Notice that r there is squared not multiplied by 2.
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A 10-by-6 inch piece of paper is used to form the lateral surface of a 11 Nov 2017, 08:08
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OCDianaOC
V = pi(5/pi)*2 x 6 = 150/pi. I'm confused as too why two pi turns into 150/pi.

V = pi(3/pi)*2 x 10 = 90/pi. I'm confused as too why two pi turns into 90/pi.

Can someone explain? Aren't we supposed to cross-cancel the pi out?

The volume formula is \(volume=\pi{r^2}h\). Notice that r there is squared not multiplied by 2.
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I'm aware of that. Sorry, I put "*" hoping to imply it meant squared since I couldn't find another symbol for that on my keyboard. Let me rephrase,

V = pi(5/pi)^2 x 6 = 150/pi. I'm confused as too why two pi turns into 150/pi.

V = pi(3/pi)^2 x 10 = 90/pi. I'm confused as too why two pi turns into 90/pi.

Can someone explain? Aren't we supposed to cross-cancel the pi out?
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A 10-by-6 inch piece of paper is used to form the lateral surface of a 11 Nov 2017, 08:15
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OCDianaOC
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V = pi(5/pi)*2 x 6 = 150/pi. I'm confused as too why two pi turns into 150/pi.

V = pi(3/pi)*2 x 10 = 90/pi. I'm confused as too why two pi turns into 90/pi.

Can someone explain? Aren't we supposed to cross-cancel the pi out?

The volume formula is \(volume=\pi{r^2}h\). Notice that r there is squared not multiplied by 2.

I'm aware of that. Sorry, I put "*" hoping to imply it meant squared since I couldn't find another symbol for that on my keyboard. Let me rephrase,

V = pi(5/pi)^2 x 6 = 150/pi. I'm confused as too why two pi turns into 150/pi.

V = pi(3/pi)^2 x 10 = 90/pi. I'm confused as too why two pi turns into 90/pi.

Can someone explain? Aren't we supposed to cross-cancel the pi out?
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\(\pi\) there is also gets squared.

h = 6 and \(r=\frac{5}{\pi}\):

\(volume=\pi{r^2}h=\pi*(\frac{5}{\pi})^2*6=\pi*\frac{25}{\pi^2}*6=\frac{150}{\pi}\).
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A 10-by-6 inch piece of paper is used to form the lateral surface of a 11 Nov 2017, 09:12
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Oh! That's what I was missing! Thanks Brunel... so we multiple out straight across (squaring num and denom) then cancel out the extra pi from numerator and denominator! :)
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A 10-by-6 inch piece of paper is used to form the lateral surface of a 02 Mar 2018, 11:08
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Hi All,

This question ultimately comes down to a couple of geometry formulas:

Circumference = 2pi(radius)
Volume = (Height)pi(radius)^2

The "lateral surface of a cylinder" is a fancy way of staying "the outside of the can, but not the top nor the bottom." With a 10x6 piece of paper, we can have 2 possible cylinders:

1) Height of 10, Circumference of 6
2) Height of 6, Circumference of 10

The answers ask us to consider the volumes of the cylinders…..

1st cylinder:
Circumference = 6 = 2pi(radius)
Radius = 3/pi

Height = 10
Volume = 10pi(3/pi)^2 = 90/pi

2nd cylinder:
Circumference = 10 = 2pi(radius)
Radius = 5/pi

Height = 6
Volume = 6pi(5/pi)^2 = 150/pi

So the cylinder with a height of 6 has a volume that is 60/pi greater than the cylinder with a height of 10.

Final Answer:
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B


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