Results 1 to 4 of 4

Math Help - Optimization problem

  1. #1
    Member
    Joined
    Jul 2007
    Posts
    147

    Optimization problem

    Dear forum members,

    in a problem I was asked to show that when the surface area of a fixed cylinder was at its minimum, the height of the cylinder would be equal to the diameter of the cross section of the cylinder.

    I formed the equation for the surface area of the cylinder, and the differentiated it with respect to r(not sure why I choose r, though. Perhaps because it was the only factor that would influence both the area of the "bottoms" of the cylinder as well as the "body".).

    A(r)=2pi*r^2 + 2 pi*r*h

    A'(r)=4pi*r+2pi*h

    solving for the stationary points of the derivative gives 2r=-h .
    The result shows basically what I wanted to show, but the minus is probably not supposed to be there.

    However, I noticed, that when I expressed h in terms of the volume and the radius

    h=V/pi*r^2

    and then plugged that into the equation above and differentiated, the result was 2r=h.

    Could someone explain to me why these two different ways of doing give a different result?

    All help is appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,621
    Thanks
    426
    Quote Originally Posted by Coach View Post
    Dear forum members,

    in a problem I was asked to show that when the surface area of a fixed cylinder was at its minimum, the height of the cylinder would be equal to the diameter of the cross section of the cylinder.

    I formed the equation for the surface area of the cylinder, and the differentiated it with respect to r(not sure why I choose r, though. Perhaps because it was the only factor that would influence both the area of the "bottoms" of the cylinder as well as the "body".).

    A(r)=2pi*r^2 + 2 pi*r*h

    A'(r)=4pi*r+2pi*h ... this derivative is incorrect, you cannot treat h as a constant.

    solving for the stationary points of the derivative gives 2r=-h .
    The result shows basically what I wanted to show, but the minus is probably not supposed to be there.

    However, I noticed, that when I expressed h in terms of the volume and the radius

    h=V/pi*r^2

    and then plugged that into the equation above and differentiated, the result was 2r=h.

    ... this is correct because V is a constant, and the variation of h is related to the variation of r.

    Could someone explain to me why these two different ways of doing give a different result?
    .
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jul 2007
    Posts
    147
    Thank you so much for your reply!

    But why can't I treat h as a constant?(I'm sorry, this is a reall stupid question, but I don't understand why is it incorrect to treat h as a constant)?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,621
    Thanks
    426
    Quote Originally Posted by Coach View Post
    Thank you so much for your reply!

    But why can't I treat h as a constant?(I'm sorry, this is a reall stupid question, but I don't understand why is it incorrect to treat h as a constant)?
    V = \pi r^2 h

    if V is constant and r changed, would h stay the same?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] optimization problem
    Posted in the Calculus Forum
    Replies: 12
    Last Post: October 6th 2011, 03:47 PM
  2. Optimization problem
    Posted in the Advanced Applied Math Forum
    Replies: 3
    Last Post: March 4th 2011, 10:58 PM
  3. Optimization Problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 20th 2009, 07:41 PM
  4. help!!!- optimization problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 13th 2008, 04:59 PM
  5. Optimization Problem
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: April 8th 2008, 05:55 PM

Search Tags


/mathhelpforum @mathhelpforum