Results 1 to 3 of 3

Math Help - Optimization

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    16

    Optimization

    A cone is cut from a sphere of radius a. You are to find the cone with the largest volume. Calculate the volume of this cone exactly.

    Let variable r be the radius of the circular base and variable h the height of the inscribed cone as shown in the two-dimensional side view.

    Optimization-1.jpg
    It is given that the circle's radius is a. Let variable z be the height of the small right triangle.
    Optimization-2.jpg
    By the Pythagorean Theorem it follows that
    r^2 + z^2 = a^2
    so that
    z^2 = a^2 - r^2
    or

    z=sqrt(a^2 - r^2)

    Thus the height of the inscribed cone is

    h = a + z = a + sqrt(a^2 - r^2)

    We wish to MINIMIZE the total VOLUME of the CONE

    V = (1/3)πr^2h

    Sub in h
    V = (1/3)πr^2(a + sqrt(a^2 - r^2))

    This is where I get stuck because I do not know where to go from here.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by mortalcyrax View Post
    A cone is cut from a sphere of radius a. You are to find the cone with the largest volume. Calculate the volume of this cone exactly.

    Let variable r be the radius of the circular base and variable h the height of the inscribed cone as shown in the two-dimensional side view.

    Click image for larger version. 

Name:	1.jpg 
Views:	128 
Size:	19.6 KB 
ID:	18508
    It is given that the circle's radius is a. Let variable z be the height of the small right triangle.
    Click image for larger version. 

Name:	2.jpg 
Views:	288 
Size:	20.8 KB 
ID:	18509
    By the Pythagorean Theorem it follows that
    r^2 + z^2 = a^2
    so that
    z^2 = a^2 - r^2
    or

    z=sqrt(a^2 - r^2)

    Thus the height of the inscribed cone is

    h = a + z = a + sqrt(a^2 - r^2)

    We wish to MINIMIZE the total VOLUME of the CONE

    V = (1/3)πr^2h

    Sub in h
    V = (1/3)πr^2(a + sqrt(a^2 - r^2))

    This is where I get stuck because I do not know where to go from here.
    Use the intersecting chord theorem to relate the radius of the base of the cone to its height and the radius of the sphere. Then substitute either the radius of the base of the height of the cone from that relation into the formula for the volume of the cone. Then find the minimum volume of the cone.

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    16
    Problem has been solved
    Last edited by mortalcyrax; August 13th 2010 at 08:24 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. optimization help!
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 12th 2009, 12:54 AM
  2. Optimization
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 8th 2009, 02:09 PM
  3. Optimization
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 29th 2009, 10:56 AM
  4. optimization
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 12th 2008, 10:47 AM
  5. Optimization
    Posted in the Pre-Calculus Forum
    Replies: 0
    Last Post: October 13th 2008, 06:44 PM

/mathhelpforum @mathhelpforum