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Fixed dimensions?

  1. May 22, 2007 #1
    1. The problem statement, all variables and given/known data
    Explain why, using this outline for creating the cylinder, the volume of the cylinder is fixed for an A4 piece of paper (30x21)

    2. Relevant equations

    I really dont know how to approach this question, i know it has something to do with the radius and the height of the cylinder. but i dont know how to prove it.
  2. jcsd
  3. May 22, 2007 #2
    The paper has certain dimensions, height = 21, width = 30.

    A cylinder has a certain h, with circles of radius r at the base. Thus, if the radius of the circles are r, the width of the paper (30) must equal:

    4*r + h = 30

    (4 radii from two circles plus the height of the cylinder = width of the paper)

    However the height of the paper must be equal to the circumference of one of the circles (2*pi*r) in order to form a cylinder:

    2*pi*r = 21

    thus r = (21/2*pi) or approximately 3.34 (cm)

    giving us the value for h = 30 - 4*(21/2*pi). or approximately 16.63 (cm)

    Making a cylinder with bases of diameter 6.68 cm and a height of 16.63 cm. and a volume of 2*pi*r^2*h = 2*3.14159*3.34^2*16.63

    = 1165.64 cm^3

    Which is the answer to the problem.

    === Warning, thinking outside the box:

    However one can argue that one could put the height of the cylinder along the height of the paper, in which case:

    4*r + 2*pi*r = 30 and (edit: that is, 4 radii of two circles plus the circumference of the cylinder's base = width of the paper)
    h = 21

    Which gives us as solutions:

    r = 30/(4+2*pi) = approximately 2.92 cm
    h = 21 cm

    And a volume of 2*pi*r^2*h = 1125.03 cm^3

    So theoretically there are two solutions to the problem. (but in any case the Volume of the cylinder is FIXED, that is a certain value).

    JJ +
    Last edited: May 22, 2007
  4. May 22, 2007 #3
    Bonus Question

    Bonus Question !

    Here is a bonus question that I thought of while looking at your question. It actually has an interesting answer:

  5. May 22, 2007 #4
    The maximum volume is:

    607.3 cm^3
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