An open box is to be made from a 21 ft by 56 ft rectangular piece of sheet metal by cutting out squares of equal size from the four corners and bending up the sides. Find the maximum volume that the box can have.

Answer 1

Let x be the length of the side of the square taken off each corner. The open box will have length 56-2x, width 21-2x and height x.

The volume of the box is
V(x) = x(56-2x)(21-2x)

Set the derivative dV/dx equal to zero and compute the corresponding x. Use that to compute the maximum volume.

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