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Homework Help: Finding min. value of dimensions for a tank of water

  1. Nov 27, 2013 #1

    1. The problem statement, all variables and given/known data

    An open tank is constructed with a square base and vertical sides to hold 32m3 of water. Find dimensions of the tank if the area of the sheet metal used to make it is to have a minimum value.

    2. Relevant equations

    3. The attempt at a solution

    I'm not entirely sure of how to approach this problem beyond needing to use the second derivative. I think I need to construct an expression of the area related to volume.


    the sides pf the base, x multiply together to give an area x2 and the four sides can be called 4xh (side of the base multiplied by height)

    x2 + 4xh is an area of sheet metal

    x2h = 32

    I don't know how to proceed.

    Thank you for any help you can offer
  2. jcsd
  3. Nov 27, 2013 #2


    Staff: Mentor

    The only thing you're missing is that your volume function can be solved for h as a function of x, and then substituted into your surface area function, making it a function of x alone. Once you have the surface area as a function of x, use the usual technique for finding the minimum value.
  4. Nov 27, 2013 #3
    ah of course! Thanks
  5. Nov 27, 2013 #4

    Ray Vickson

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    Besides the hint already given, you can also use the Lagrange multiplier method (if you know about it).
  6. Nov 28, 2013 #5
    I don't know about that.

    I'm in a physics foundation year (year 0 at my uni) due to not having done the traditional subjects that lead to taking a physics degree.
  7. Nov 28, 2013 #6


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    There is a shortcut available here. If the tank also had a top, you'd straight away say it should be a cube. With the open tank, imagine creating a closed tank by taking two optimised open tanks and inverting one over the other. You must now have created an optimal closed tank, i.e. a cube.
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