# Squares are cut from each corner of a #0.5m xx 0.5m# square of cardboard and the sides folded up to make an open box. What is the maximum possible volume of the box?

##### 1 Answer

Consider the general case of a square sheet of cardboard with sides of length

The area of the bottom of the resulting box will be

The height of the box will be

The volume of the box will be

The maximum of this will occur at some point where the derivative is zero...

So

If

So the maximum volume occurs when

Then

If

and