Both the profit a company gets per item, P, and the quantity of an item it sells, q, will depend upon the selling price, p.

Question

Both the profit a company gets per item, P, and the quantity of an item it sells, q, will depend upon the selling price, p. Suppose when p = $12 a company sells q = 250 items with a profit of P = $2 each, and at this price, q is decreasing at a rate of 10 items per dollar increase in price, while P is increasing at a rate of $0.80 dollars profit/item per dollar increase in price. At what rate is the company’s total profit for the product changing with respect to the selling price?

Answer 1

Here's a calculus based solution.

P(p) as Profit depends on price

Q(p) as Demand depends on price


Q(12) = 250
P(12) = 2

Q'(12) = -10
P'(12) = 0.80

The first goal is to find P(p)

y = Mx + b

P(p) = dP/dp * x + b

2 = (0.80)(12) + b

2 = 9.6 + b

b = -7.6

P(p) = (0.80)p - 7.6


The next goal is to find Q(p)

y = Mx + b

Q(p) = dQ/dp * x + b

250 = (-10)(12) + b

250 = -120 + b

b = 370

Q(p) = -10p + 370


The revenue function R(p) = p*Q(p) = p ( -10p + 370)

= -10p^2 + 370p

dR/dp = -20p + 370 = R'(p)