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Question

A box with a hinged lid is to be made out of a piece of cardboard that measures 30 by 40 inches. Six squares, x inches on a side, will be cut from each corner and the middle of the long sides, and then the ends and sides will be folded up to form the box and its lid.

What dimensions will give the maximum volume for the box?

Solution

Let the side size of the square cut out of each corner be \"a\".

Then volume of box = V = (30-2a) (40-2a) a

= a(4a² - 140a + 1200)

dV/da = 12a² -280+1200 = 0 or a²-35a+300 =0 for a maximum.

Hence:
a=[35±(1225-1200)]/2 giving a = [35±15]/2 = either 25 or 5

Obviously only a = 5 inches is viable.
Size of box base = 25 35 inches
Height of the box = 5 inches.