A rain gutter is to be made up of rectangular aluminum sheets 12 inches wide by turning up the sides edges 90 degrees. What depth (of the edges) will provide a maximum cross sectional area and thereby provide for the greatest flow of water? (Hint, think of quadratic equations and you want to maximize the area.)

and

The marketing department at Texas Instruments has found that when certain calculators are sold at a price of p dollars, the revenue (in dollars) as a function of the price is:
R(p) = -150p2 + 21,000p
(a) What unit price should be established in order to maximize revenue?
(b) If this price is charged what is the maximum revenue.
(c) Graph the function R(p).

The sheeting is 12" wide. Turn up two edges of length x.

Therefore, the bottom will have width 12-2x and the height is x.

The area is

This is a quadratic. The max is at the vertex, which can be found by

.