Aone 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

h = 2.5 in

The vertex of the quadratic function is (1.5, -.75)