A 10-by-6 inch piece of paper is used to form the lateral surface of a

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)

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.

HI Bunuel,

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

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?

\(\pi\) there is also gets squared.

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

\(volume=\pi{r^2}h=\pi*(\frac{5}{\pi})^2*6=\pi*\frac{25}{\pi^2}*6=\frac{150}{\pi}\).

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!

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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