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Math Help - volume

  1. #1
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    volume

    A spherical cap of radius p and height h is cut from a sphere of radius r. Show that the volume V of the spherical cap can be expressed as

    A) (1/3)pi h^2 (3r-h)
    B) (1/6)pi h (3p^2 + h^2)

    Thanks in advance
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  2. #2
    Bar0n janvdl's Avatar
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    Could you perhaps illustrate with a picture what you mean?
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by janvdl View Post
    Could you perhaps illustrate with a picture what you mean?
    This is what he means:
    Attached Thumbnails Attached Thumbnails volume-gash.jpg  
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  4. #4
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    Hello, Csou090490!

    I can help with part (A) . . .


    A spherical cap of radius p and height h is cut from a sphere of radius r.
    Show that the volume V of the spherical cap can be expressed as:

    A) (1/3)πh(3r - h)

    B) (1/6)π h(3p + h)
    Code:
                    |
                  * * *
              *     |     *
            *       |     |:*
           *        |     |::*
                    |     |:::
          *         |     |:h:*
        --*---------+-----+---*--
          *         |    r-h  * r
                    |
           *        |        *
            *       |       *
              *     |     *
                  * * *
                    |
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ._____
    We have a circle: .x + y .= .r . . y .= .√r - x

    The shaded region indicated above is revolved about the x-axis.

    Its volume is: .V .= .π ∫ (r - x) dx . . . from .x = r-h .to .x = r

    . . V .= .π[rx - x/3] . . . x = r-h to r

    . . . .= .π
    ([r - r/3] - [r(r-h) - (r-h)/3])

    which simplifies to: .(1/3)πh(3r - h)

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