A one compartment vertical We is to be constructed by bendin

A one compartment vertical We is to be constructed by bending the long aide of an 8 in. by 12 in. sheet of plastic along two lines to term a U shape. How tall should the file bo to maximize the volume that it can hold?

Solution

The sides of the drawer are formed by the long side of the sheet, so the depth of the drawer will be 8 in.
Since the depth of the drawer is fixed, it will play no role in the maximization problem.
So we need to maximize the area of the drawer formed by the two sides and bottom.
Let w = the width of the drawer
Let h = the height of the drawer
Then w + 2h = 12.......... [1]

The area, A = w*h.......... [2]
Solve for w in [1] and substitute into [2]
w = 12 - 2h
A = (12-2h)h
To maximize the area compute dA/dh and set=0:
dA/dh = 0 = 12 - 4h
Solving for h gives h = 3
Therefore, the height is 3 in.