Question

If you roll an 8.5-by-11-inch piece of paper into a cylinder by bringing the two
longer sides together, you get a tall, thin cylinder. If you roll an 8.5-by-11-inch piece
of paper into a cylinder by bringing the two shorter sides together, you get a short,
fat cylinder. Which of the two cylinders has the greater volume?

Answer 1

visualize the problem, find out the perimeter of the circle formed by joining the two edges in both cases. you will get the radius that way.

Answer 2

Do you know the formula in finding the volume of cylinder?
Its the B * H but in this case the B is circular so its the area of a circle
Pi r^2 * H
you know if you roll it the long way you have a height of 11 and if you roll it the other way you have a H of 8.5

so no you have
Pi r^2 * 11= Volume in the long way
Pi r^2 * 8.5= Volume the short way

Now you need to find the radius. Can you think of a way in finding the radius?

Answer 3

so B=pir3 right

Answer 4

no the base is\[B=\pi r^2\]

Answer 5

my bad what is radius if this

Answer 6

Lets say you roll it up the long way with the H=11
when you roll it you can see the base right? Can you make a connection between the base and the width of the paper (8.5)?

Answer 7

that mean 4.25

Answer 8

wait how did you get 4.25?

Answer 9

Here is a clue when you roll it the circumference ( 2 pi r) is 8.5 so no solve for r.

Answer 10

now solve for r

Answer 11

\[2\pi r=8.5\]

Answer 12

now that you have that and now you can figure out what r is plug it back to the volume equation and thats the first volume

Answer 13

Next solve for the second volume. The circumference is going to be different.

Answer 14

its not going to be 8.5 but what?

Answer 15

\[2pir=8.5\]what that for

Answer 16

You know what's the circumference right? the length around a circle. Because you rolled up the paper the length of 8.5 becomes the circumference

the formula for the circumference is

2 Pi r
and you know that the circumference is 8.5 so

\[2 \pi r = 8.5\]

Answer 17

solve for r and that the radius of the circle or the base. But you want the area of the base so plug that in to the original Volume formula got it?

Answer 18

so that divde by 2pi