An open box with a square base is to be made out of a given quantity of cardboard of area c² square units. Show that the maximum volume of the box is c³/6√3 cubic units.

The area of the square piece is given to be c² square units.
Since the base is a square, let the length and breadth of the resulting box be l and the height be h.
Therefore, l² + 4lh = c², equating the areas.
Also, the volume of the box is thus given by length × breadth × height = l²h
We can write the volume only in terms of l as l² × (c² - l²) / 4l = lc²/4 - l³/4
Differentiating this w.r.t l and equating it to zero, we get
c²/4 - 3l²/4 = 0
⇒ l = √(c²/3)
The volume thus becomes lc²/4 - l³/4 = √(c²/3) × c²/4 - (c²/3)³/4 = c√3 × c²/12 = c³/6√3