Results 1 to 2 of 2

Math Help - a sum dealing with cone, sphere...help!

  1. #1
    Member
    Joined
    Mar 2009
    From
    jharkhand
    Posts
    77

    a sum dealing with cone, sphere...help!

    A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed. What fraction of water will overflow?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,682
    Thanks
    614
    Hello, snigdha!

    A conical vessel of radius 6 cm and height 8 cm is completely filled with water.
    A sphere is lowered into the water and its size is such that when it touches the sides,
    it is just immersed. .What fraction of water will overflow?
    Code:
          A      6        P       6       B
        - o - - - - - - * o * - - - - - - o
          :\        *     |     *        /
          : \     *       |r      *     / 
          :  \   *        |        *   /
          :   \           |           /
          :    \*         oO        */ 6
          :     \         | *  r    / 
          :      *        |   *    *
        8 :       *       |r    * *
          :        *      |      o Q
          :         \   * o *   /
          :          \    |    /
          :           \   |   /
          :            \  |  / 4
          :             \ | /
          :              \|/
          -               o
                          C

    ABC is the cone, with altitude H \,=\,PC \,=\, 8
    Its radius is: . R \,=\,AP\,=\,PB\,=\,6

    In right triangle BPC, Pythagorus tell us: . BC \,=\,10

    Note that: . BQ \,=\,BP \,=\,6. . Hence: . QC \,=\,4


    The center of the sphere is O with radius r\!:\;\;OP \,=\, OQ \,=\, r


    In right triangle OQC\!:\;\;OQ^2 + QC^2 \:=\:OC^2

    We have: . OQ \,=\,r,\;QC\,=\,4,\;OC \,=\,8-r

    Then: . r^2 + 4^2 \:=\:(8-r)^2 \quad\Rightarrow\quad 16r \:=\:48 \quad\Rightarrow\quad\boxed{ r \:=\:3}


    The volume of the sphere is: . V_s \:=\:\tfrac{4}{3}\pi r^3 \:=\:\tfrac{4}{3}\pi (3^3) \:=\:36\pi
    . . This is the volume of water that will overflow.

    The volume of the cone is: . V_c\:=\:\tfrac{1}{3}R^2H \:=\:\tfrac{1}{3}\pi(6^2)(8) \:=\:96\pi


    The fraction is: . \frac{36\pi}{96\pi} \:=\:\frac{3}{8}


    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. sphere cone problem
    Posted in the Geometry Forum
    Replies: 1
    Last Post: January 31st 2010, 10:18 AM
  2. Sphere in Cone Geometry
    Posted in the Geometry Forum
    Replies: 2
    Last Post: March 16th 2009, 07:08 AM
  3. sphere and cone
    Posted in the Geometry Forum
    Replies: 1
    Last Post: December 22nd 2008, 08:59 PM
  4. integral (within sphere, below cone)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 31st 2008, 06:52 AM
  5. Sphere into cone
    Posted in the Geometry Forum
    Replies: 13
    Last Post: January 17th 2006, 08:35 PM

Search Tags


/mathhelpforum @mathhelpforum