Results 1 to 2 of 2

Math Help - A cylindrical frame volume max

  1. #1
    Newbie
    Joined
    Dec 2006
    Posts
    10

    A cylindrical frame volume max

    A cylindrical kite frame is to be constructed from a 4-m length of light bendable rod.

    The frame will be made up of 2 circles joined by four straight rods of equal length. In order to maximize lift, the kite frame must be constructed to maximize the volume of the cylinder.

    In what lengths should the peices be cut to optimize the kites flight?

    I know the answer but every time I try to do it, I get a different one, help would be appreicated greatly.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by Sky12 View Post
    A cylindrical kite frame is to be constructed from a 4-m length of light bendable rod.

    The frame will be made up of 2 circles joined by four straight rods of equal length. In order to maximize lift, the kite frame must be constructed to maximize the volume of the cylinder.

    In what lengths should the peices be cut to optimize the kites flight?

    I know the answer but every time I try to do it, I get a different one, help would be appreicated greatly.
    Hello,

    The volume of a cylindre is calculated by:

    V=\pi \cdot r^2 \cdot h

    You've got the additional condition:

    4 = \underbrace{2 \cdot 2 \pi r}_{\text{two circles}}+4h
    Solve this equation for h:

    h=1-\pi r. Now plug in this term for h into the first equation and expand:

    V=\pi \cdot r^2 \cdot (1-\pi r)=\pi r^2- \pi^2 \cdot r^3

    The volume has an extreme value if the first derivative equals zero:

    V'(r)=2 \pi r-3 \pi^2 r^2=\pi r(2-3\pi r)

    Now solve the equation V'(r) = 0 for r. You'll get r = 0. This kite has the volume zero, which is indeed an extremum. The 2nd value for r is:
    r=\frac{2}{3 \pi}\ \Longrightarrow \ h=\frac{1}{3}

    The maximal volume is V_{max}=0.047 \text{ m}=12.4\text{ gal(US)}

    EB
    Attached Thumbnails Attached Thumbnails A cylindrical frame volume max-zyl_abmess1.gif  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Volume for a cone in cylindrical coordinates.
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 1st 2011, 05:44 AM
  2. Volume of a cylindrical shell
    Posted in the Calculus Forum
    Replies: 9
    Last Post: October 7th 2010, 01:06 PM
  3. Finding the volume using cylindrical shells
    Posted in the Calculus Forum
    Replies: 0
    Last Post: August 25th 2010, 04:24 AM
  4. Volume by cylindrical shells
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 3rd 2010, 09:56 PM
  5. Volume by cylindrical shells
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 9th 2008, 03:42 PM

Search Tags


/mathhelpforum @mathhelpforum