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Thread: How do you get a Cylinder out of a Sphere?

  1. #1
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    How do you get a Cylinder out of a Sphere?

    Hi folks,

    A cylinder of volume V is to be cut from a solid sphere of radius R. Prove that the maximum value of V is $ \frac{4 \pi R^3}{3 \sqrt{3}}$

    Volume of the cylinder V = $\pi r^2 h$

    From the attached drawing $R^2 = r^2 + \frac{h^2}{4}$

    making h the subject of this equation, we get $h = 2 \sqrt{R^2 - r^2}$ and substitute back

    V = $2 \pi r^3 (\frac{R^2}{r^2} - 1)^\frac{1}{2}$

    The idea is to differentiate V with respect to r, set to 0 (for a max) and get the terms in r to drop out, but the expression is quite a handful and I suspect I am on the wrong track.

    Any advice?
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  2. #2
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    Re: How do you get a Cylinder out of a Sphere?

    reference the attached diagram ...

    $V = \pi x^2 \cdot 2y$

    $x^2 = R^2-y^2 \implies V = 2\pi y(R^2-y^2) = 2\pi(R^2 y - y^3)$

    $\dfrac{dV}{dy} = 2\pi(R^2 - 3y^2) = 2\pi[R^2 - 3(R^2 - x^2)] = 2\pi(3x^2 -2R^2)$

    $\dfrac{dV}{dy} = 2\pi(3x^2 -2R^2) = 0 \implies x^2 = \dfrac{2R^2}{3}$



    $V = 2\pi x^2 y = 2\pi \cdot \dfrac{2R^2}{3} \sqrt{R^2-x^2} = \dfrac{4\pi R^2}{3} \sqrt{\dfrac{R^2}{3}} = \dfrac{4\pi R^3}{3 \sqrt{3}}$
    Attached Thumbnails Attached Thumbnails How do you get a Cylinder out of a Sphere?-cylinder_v.jpg  
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    Re: How do you get a Cylinder out of a Sphere?

    Thanks Skeeter. Seems I was on the right track, but too messy! Your solution is nice and neat.
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    Re: How do you get a Cylinder out of a Sphere?

    Sorry Skeeter, I'm back again.

    Actually, your solution isn't just neat; you decided to differentiate with respect to the cylinder height rather than the cylinder radius. We both started with $V = \pi r^2 h$ and we both had the same relationship between cylinder height, cylinder radius and sphere radius. They key thing was your decision to work with the height. In principle it should work both ways, but if you do what I did, go with the radius it gets very horrid and I don't think one can avoid this. Did you have a reason for selecting the height or did it just pan out that way? What I mean is, did you see the way I did it, see that it was a disaster and just try it the other way, or was their some insight that you had?
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    Re: How do you get a Cylinder out of a Sphere?

    With regard to derivatives, I try to avoid radicals if possible ... sometimes you can't. In this case, it was doable.
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