An open rectangular box having a volume of 108 in.3 is to be constructed from a tin sheet. find the dimensions of such a box if the amount of material used in its construction is to be minimal. hint: let the dimensions of the box be x in. by y in. by z in. then, xyz = 108 and the amount of material used is given by

It works out the the optimum box dimensions are those that make the lateral area twice the base area. Of course, the base is a square in order to minimize the lateral area for a given base area.

That being the case, the height of the box is half the base edge length and the dimensions are
.. x = y = 6 in
.. z = 3 in

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If x and y are the base dimensions, the volume is
.. xyz = 108
and we want to minimize the area
.. area = xy +2(x+y)z

These equations are symmetrical in x and y, so the solution will require x=y. Making that substitution, we have
.. x^2*z = 108
.. x^2 +4xz = area = x^2 +4x(108/x^2)

That is, we want to minimize x^2 +432/x. A graphing calculator finds the minimum at x=y=6, (and z = 108/x^2 = 3).