Results 1 to 3 of 3

Math Help - cylinder n sphere

  1. #1
    Senior Member furor celtica's Avatar
    Joined
    May 2009
    Posts
    271

    cylinder n sphere

    a circular cylinder is to fit inside a sphere of radius 10cm. calculate the maximum possible volume of the cylinder. (it is probably best to take as your indpendent variable the height or half the height of the cylinder).
    i'm stuck here cos i don't know where to start. i drew a section of the both to try and find a clue but i'm lost. any hints?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,707
    Thanks
    627
    Hello, furor celtica!

    A circular cylinder is inscribed in a sphere of radius 10 cm.
    Calculate the maximum possible volume of the cylinder.
    Code:
                  * * *
              *           *
            * - - - + - - - *
           *|       |       |*
            |       |       |
          * |       |       | *
          * |       *       | *
          * |       | * 10  | *
            |      y|   *   |
           *|       |  x  * |*
            * - - - + - - - *
              *           *
                  * * *

    The radius of the cylinder is x.
    The height of the cylinder is 2y.

    We see that: . x^2+y^2\:=\:10^2 \quad\Rightarrow\quad y \:=\:\sqrt{100-x^2}


    The volume of the cylinder is: . V \;=\;\pi r^2h

    So we have: . V \;=\;\pi x^2(2y)

    . . . . . . . . . . V \;=\;2\pi x^2(100-x^2)^{\frac{1}{2}}


    And that is the function we must maximize.

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member AllanCuz's Avatar
    Joined
    Apr 2010
    From
    Canada
    Posts
    384
    Thanks
    4
    Quote Originally Posted by Soroban View Post
    Hello, furor celtica!


    Code:
                  * * *
              *           *
            * - - - + - - - *
           *|       |       |*
            |       |       |
          * |       |       | *
          * |       *       | *
          * |       | * 10  | *
            |      y|   *   |
           *|       |  x  * |*
            * - - - + - - - *
              *           *
                  * * *
    The radius of the cylinder is x.
    The height of the cylinder is 2y.

    We see that: . x^2+y^2\:=\:10^2 \quad\Rightarrow\quad y \:=\:\sqrt{100-x^2}


    The volume of the cylinder is: . V \;=\;\pi r^2h

    So we have: . V \;=\;\pi x^2(2y)

    . . . . . . . . . . V \;=\;2\pi x^2(100-x^2)^{\frac{1}{2}}


    And that is the function we must maximize.
    You know Soroban...I saw this topic and got all giddy when there were no responses. Now that I am no longer the first, I no longer feel so giddy! Thanks for taking that away lol

    I would solve it a little bit differently.

    Equation of the sphere  z^2 + x^2 + y^2 = 10^2

    Equation of the cylinder  x^2 + y^2 = a^2

    Subbing the cylinder equation into the sphere equation,

     z^2 + a^2 = 100 \to z= \sqrt{ 100 -a^2 }

    Transforming to cylindrical co-ordinates,

     V = 2 \int_0^{ 2 \pi } d \theta \int_0^a rdr \int_0^{ \sqrt{100-a^2} } dz

    Compute the above and find  \frac{dV}{dA} = 0

    We would find this to be the same equation as that of Soroban, but I always like to derive max volume questions via the triple integral representation. It's good practice I feel.
    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 in a sphere
    Posted in the Geometry Forum
    Replies: 2
    Last Post: June 23rd 2009, 12:19 AM
  3. Cylinder in a Sphere
    Posted in the Geometry Forum
    Replies: 5
    Last Post: May 10th 2008, 04:16 PM
  4. Cylinder and Sphere
    Posted in the Geometry Forum
    Replies: 20
    Last Post: March 2nd 2008, 05:21 PM
  5. Cylinder/Sphere
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 6th 2007, 08:01 PM

Search Tags


/mathhelpforum @mathhelpforum