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Math Help - cylinder inscribed in sphere with given equation

  1. #1
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    cylinder inscribed in sphere with given equation

    Hi guys, i'm in some trouble right now so anyway, this is what the question is.
    Use the method of Lagrange multipliers to find the cylinder of maximum volume that can be inscribed inside an AFL style mathematical football with the geometric equation, f(x,y,z) = 15x^2 + 15y^2 + 4z^2 = 15.
    without lose of generality, let the axis of the cylinder be the z axis.

    a) Using r^2 = x^2 + y^2 , restate f(x,y,z) = 15 as g(r,z) = 15.
    b) show that the volume of an inscribed cylinder , V(r,z) = 2 pi R^2 z.
    c) Set-up and solve the Lagrange multiplier equations that maximises V(r,z) subject to g(r,z) = 15.
    d) state the exact and approximate numberical values of (i) radius, (ii) height, (iii) volume of the inscribed cylinder of maximum volume.

    SO,
    does "let the axis of the cylinder be the z axis" mean that z would be the height?
    if it isn't, what is it??
    I know my function to be maximised is V = pi r^2 h,
    and my working for part a) is,
    R^2 + (h/2)^2 - r^2 = 15 (R being the radius of the cylinder and r being the radius of the AFL mathematical football)
    i've managed to work it out by setting R^2 as the subject and plugging it back into my volume equation, but since this part i seem to be lost.
    I also don't fully understand the question, can you please explain it to me in lame-man terms? (sorry)
    am I right to set my part (a) equation to 15?
    cause if I am right, i can't seem to prove part (b) , and of course can't move on.
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: cylinder inscribed in sphere with given equation

    It appears that the largest volume cylinder that can fit inside a football has :

    V_{\max }=\frac{\sqrt{5} \pi }{3}

    for: \left\{x\to \frac{7}{16},y\to \frac{\sqrt{\frac{365}{3}}}{16},z\to \frac{\sqrt{5}}{2}\right\}

    [not double checked]
    Last edited by MaxJasper; October 14th 2012 at 10:03 AM.
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