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    dks2114's Avatar
    dks2114 Posts: 32, Reputation: 2
    Junior Member
    		  
    #1

    Jul 25, 2007, 07:45 AM
    Optimizing a Box
    Yet another optimization problem... if anyone knows how to solve this please explain it to me:

    Thank you.A box with a square base and open top must have a volume of 32,000cm3. Find the dimensions of the box that minimize the amount of material used.
    ebaines's Avatar
    ebaines Posts: 12,131, Reputation: 1307
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    #2

    Jul 25, 2007, 10:15 AM
    Set up an equation for the amount of material used as a function of one of the dimensions of the box. If H is the height and L the dimension of the square base, then you know that:

    32000 = HL^2

    The area of material used is the sum of the areas of the four sides plus the top & bottom: A = 4HL + 2*L^2.

    You can combine these to get an equation that gives A as a function of L. Then use the technique I described in response to the other question you posted to solve for the value for L that gives a minimum value for A.
    galactus's Avatar
    galactus Posts: 2,271, Reputation: 282
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    #3

    Jul 25, 2007, 11:14 AM
    You must minimize the surface area given the volume is

    Since there is no top, the surface area is given by

    Solve the volume equation for, say, y; Sub it into S; differentiate, set to 0 and solve for x.

    The y value will then follow.
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