anonymous
  • anonymous
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?
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
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.
anonymous
  • anonymous
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?
anonymous
  • anonymous
so B=pir3 right

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anonymous
  • anonymous
no the base is\[B=\pi r^2\]
anonymous
  • anonymous
my bad what is radius if this
anonymous
  • anonymous
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)?
anonymous
  • anonymous
that mean 4.25
anonymous
  • anonymous
wait how did you get 4.25?
anonymous
  • anonymous
Here is a clue when you roll it the circumference ( 2 pi r) is 8.5 so no solve for r.
anonymous
  • anonymous
now solve for r
anonymous
  • anonymous
\[2\pi r=8.5\]
anonymous
  • anonymous
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
anonymous
  • anonymous
Next solve for the second volume. The circumference is going to be different.
anonymous
  • anonymous
its not going to be 8.5 but what?
anonymous
  • anonymous
\[2pir=8.5\]what that for
anonymous
  • anonymous
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 \[2 \pi r = 8.5\]
anonymous
  • anonymous
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?
anonymous
  • anonymous
so that divde by 2pi

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