A one compartment vertical file is to be constructed by bending the long side of an 8 in 8 in. by 10 10 in. sheet of plastic along two lines to form a U shape. How tall should the file be to maximize the volume that it can hold?

Th long side of the plastic sheet is bent into a U shape.

Therefore the dimensions of the file holder are 8, h, and (10 - 2h) as shown in the figure.

The volume of the file holder is

V = 8*h * (10 - 2h) = 80h - 16h²

To maximize V, V' (h) = 0.

That is,

80 - 32 h = 0

h = 80/32 = 2.5 in

V'' (h) = - 32 guarantees that V is maximum because V'' <0.

Maximum V = 80 * (2.5) - 16 * (2.5) ² = 100 in³

Answer: h = 2.5 in