Results 1 to 2 of 2

Thread: minimize problem

  1. #1
    Mar 2008

    Thumbs up minimize problem

    The consultancy unit of a university is to introduce a new soft drink to be packed in aluminium cans each to hold 45cl.find the dimensions that will minimize the amount of material used in the construction of the can ,assuming the thickness of the material is uniform.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member roy_zhang's Avatar
    Mar 2008
    Since the shape of a can is a cylinder, let $\displaystyle r$ be the radius of the base and let $\displaystyle h$ be the height of the cylinder.

    From given we have $\displaystyle V=\pi r^2 h=45$, further let $\displaystyle S$ be the surface area of the cylinder. And $\displaystyle S= 2\pi r^2 + 2\pi r h$. Our goal is to minimize the surface area.

    From the given volume constraint, let's rewrite $\displaystyle h=\frac{45}{\pi r^2}$, plug into the surface formula above, we have:

    $\displaystyle S=2\pi r^2 + \frac{90}{r}$

    In order to find the minimum value of $\displaystyle S$, let's find the critical number(s) of the surface function by:

    $\displaystyle \frac{d}{dr}S = 4\pi r-90r^{-2}=0$, which gives $\displaystyle r=\sqrt[3]{\frac{90}{4\pi}}\approx 1.9276$ (please verify it is indeed a minimum)

    Then $\displaystyle h=\frac{45}{\pi r^2} \approx 3.8552 $
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Minimize
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Nov 26th 2009, 07:46 PM
  2. Minimize function
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 22nd 2009, 03:37 PM
  3. Point on the graph minimize problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Apr 16th 2009, 03:26 AM
  4. Minimize
    Posted in the Calculus Forum
    Replies: 7
    Last Post: Mar 19th 2009, 11:22 AM
  5. Minimize Expected Value
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: Feb 23rd 2009, 11:21 PM

Search Tags

/mathhelpforum @mathhelpforum