A company produces juice containers which have the shape of a box with squares on the base and the top, and a capacity of 2 litres.
[list=1] [*] Find a formula for the surface area, [math]A[/math], of the container in terms of the apothem [math]a[/math] of the container. The apothem is the segment from the centre of the square to the midpoint of one of its sides. [*] Find the apothem of the container which will minimize the cost of the material required to make the box. [*] Find the corresponding height and side of the square. [/list]