The price p and the quantity x sold of a certain product obe

The price p and the quantity x sold of a certain product obey the demand equation

x =- 8p + 80, 0 p l 10

(a) Express the revenue R as a function of x.

(b) What is the revenue if 24 units are sold?

(d) What price should the company charge to maximize revenue?

Revenue = Price*quantity

x = -8p + 80, and 0 <= P <= 10

8p = 80 - x

p = -x/8 + 10

R = p*x

R = x*(-x/8 + 10)

R = -x^2/8 + 10x

x = 24

R = -x^2/8 + 10x

R = -24^2/8 + 10*24

R = 168

In this parabolic equation

R = -x^2/8 + 10x

a = -1/8, b = 10

maximize x = -b/2a

x = -10/(2*(-1/8)) = 40

maximum revenue

R = -x^2/8 + 10x

R = -40^2/8 + 10*40 = 200

p = -x/8 + 10

so for x = 40

p = -40/8 + 10 = 5

price = 5