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Math Help - Cylinder in a Sphere

  1. #1
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    Cylinder in a Sphere

    Inscribed a right circular cylinder of height h and radius r in a sphere of fixed radius R. Express the volume V of the cylinder as a function of h.

    HINTS GIVEN:

    (1) V = pi(r^2)(h).

    (2) R = the hypotenuse and r = one of the legs of the right triangle in the cylinder.
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  2. #2
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    Quote Originally Posted by symmetry View Post
    Inscribed a right circular cylinder of height h and radius r in a sphere of fixed radius R. Express the volume V of the cylinder as a function of h....
    Hello,

    I've attached a diagram (as usual )

    You see that

    \left( \frac{1}{2}h\right)^2+r^2=R^2 \Longleftrightarrow r^2=R^2-\frac{1}{4}h^2

    Plug in this term for r² in the formula for the volume:

    V=\pi \cdot \left( R^2-\frac{1}{4}h^2 \right) \cdot h=\pi R h-\frac{1}{4} \cdot \pi \cdot h^3

    EB
    Attached Thumbnails Attached Thumbnails Cylinder in a Sphere-zyl_inkugl.gif  
    Last edited by earboth; January 24th 2007 at 11:00 AM.
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  3. #3
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    ok

    Your diagrams are amazingly wonderful!

    How do you do that?

    Thanks!
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  4. #4
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    Quote Originally Posted by symmetry View Post
    Your diagrams are amazingly wonderful!
    How do you do that?
    Thanks!
    Thanks for the ovation (please sit down if you are still standing).

    Have a look here:
    euklid dynageo homepage

    EB
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  5. #5
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    ok

    Thank for your help and for sharing your homepage link. I will look into it this weekend.
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  6. #6
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    Small Correction

    In the formula ...



    The first term should be (pi)(R^2)h

    Otherwise, I enjoyed the wonderful illustration and solution ... very eloquent!

    stevez2436
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