Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = x p(x) - C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal profit approximates the profit obtained by selling one more item given that x items have already been sold. Consider the following cost functions C and price functions p.

a. Find the profit function P.

b. Find the average profit function and marginal profit function.

c. Find the average profit and marginal profit if x = a units are sold.

d. Interpret the meaning of the values obtained in part (c).

C(x) = -0.02x² + 50x + 100, p(x) = 100 - 0.1x, a = 500