Results 1 to 4 of 4

Math Help - 3rd and final optimization problem

  1. #1
    Newbie
    Joined
    Jul 2012
    From
    Front Royal, VA
    Posts
    5

    3rd and final optimization problem

    Trying to design an aluminum can containing a volume of 2,000 cubic centimeters.

    Find the dimensions of the can that will minimize the amount of aluminum used.

    The three formulas I'm working with will obviously
    Volume: π r2 h=2000
    Vertical surface area: 2
    π r h
    Lid and base surface area:
    π r2 (x2)

    I understand that basically what I'm trying to do is isolate one variable (for example make h=2000/
    π r2) so that I can now substitute that equation in for the variable h in another formula.

    That will result in a quadratic equation, from which I will find the derivative, find max and min, and figure out the volume at the minimum. Am I missing something? I keep attempting this, but I keep getting stuck after I
    substitute 2000/πr^2 for h because I don't have a quadratic equation to work with.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member BAdhi's Avatar
    Joined
    Oct 2010
    From
    Gampaha, Sri Lanka
    Posts
    252
    Thanks
    6

    Re: 3rd and final optimization problem

    you don't always need a quadratic equation to solve a optimization problem,

    when the volume is  V,

    \begin{align*}V&=\pi r^2h\\h&=\frac{V}{\pi r^2}\\\end{align*}

    then,
    \begin{align*}A&=2\pi rh +2\pi r^2\\&=2\pi r\left[\frac{V}{\pi r^2}\right]+2\pi r^2\\ &= 2 \left[ \frac{V}{r}+\pi r^2\right] \end{align*}

    to find the maximum point you can use differentiation,

    \frac{dA}{dr}=2\left[2\pi r- \frac{V}{r^2}\right]

    equate this to zero to find an extrema.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jul 2012
    From
    Front Royal, VA
    Posts
    5

    Re: 3rd and final optimization problem

    Once again you come to the rescue! Thank you so very much for explaining things in a way that I can easily understand them. You have already helped me more than my textbook has.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member BAdhi's Avatar
    Joined
    Oct 2010
    From
    Gampaha, Sri Lanka
    Posts
    252
    Thanks
    6

    Re: 3rd and final optimization problem

    I'm glad that I could be a help
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Final Linear Programming Problem
    Posted in the Advanced Math Topics Forum
    Replies: 3
    Last Post: June 4th 2009, 07:39 AM
  2. Final sequence problem!
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 18th 2008, 11:35 AM
  3. Please Help Final Tom!! Word Problem
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: December 1st 2007, 12:24 AM
  4. Please Help Final Tom!! Word Problem
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: November 30th 2007, 08:53 PM
  5. The final problem..
    Posted in the Algebra Forum
    Replies: 4
    Last Post: November 4th 2007, 06:18 PM

Search Tags


/mathhelpforum @mathhelpforum