Results 1 to 2 of 2

Math Help - Cylinder/Sphere

  1. #1
    Member
    Joined
    Sep 2006
    Posts
    221

    Cylinder/Sphere

    A cylindrical drill, which has radius d, is used to drill a hole through the center of a sphere which has radius R. Determine the volume of the ring-shaped solid that would remain.

    Then, show that the volume will depend on only the height of the ring when its tallest.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,683
    Thanks
    615
    Hello, Ideasman!

    This is a classic problem . . .


    A cylindrical drill, which has radius d, is used to drill a hole
    through the center of a sphere which has radius R.
    Determine the volume of the ring-shaped solid that would remain.

    Then, show that the volume depends on only the height of the ring.
    Code:
                    |
                  * * *
              *:::::|:::::*
            * - - - + - - - *
           *        |  h  *  *
                   d|   * R
          *         | *       * 
      - - * - - - - + - - - - * - -
          *         |         *
                    |
           *        |        *
            *       |       *
              *     |     *
                  * * *
                    |

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .______
    We have the circle: .x + y .= .R . . y .= .√R - x

    The region cut off by y = d is revolved about the x-axis.
    The intersections are: .(h, d)
    . . Note that: .h .= .R - d

    . . . . . . . . . . . . . . . . . . . . . . . . . . .______
    The volume of the ring is: .2 π ∫ [(√R - x) - d] dx ... from 0 to h

    We have: .V .= .2π ∫(R - x - d) dx .= .2π ∫(R - d - x) dx

    Since R - d .= .h, we have: .V .= .2π ∫ (h - x) dx .= .2π(hx - x/3)

    Evaluate from 0 to h: .V .= .2π(h - h/3) .= .4πh/3

    Since h is half the height of the hole (H): .h = H

    Therefore: .V .= .4π(H/2)/3 .= .πH/6

    The volume of the ring is a function of H only.
    (The radius of the sphere and the size of the drill are irrelevant!)

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. cylinder in a sphere
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 14th 2010, 12:16 PM
  2. cylinder n sphere
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 10th 2010, 07:41 AM
  3. cylinder in a sphere
    Posted in the Geometry Forum
    Replies: 2
    Last Post: June 23rd 2009, 12:19 AM
  4. Cylinder in a Sphere
    Posted in the Geometry Forum
    Replies: 5
    Last Post: May 10th 2008, 04:16 PM
  5. Cylinder and Sphere
    Posted in the Geometry Forum
    Replies: 20
    Last Post: March 2nd 2008, 05:21 PM

Search Tags


/mathhelpforum @mathhelpforum