Question

Suppose that a company wants to choose a price for their product that will maximize the total revenue. From market research, they assume that the price of the product (p) and the quantity sold (x) will obey the linear relationship p=100-1/2x

(A) What happens if the company sets the price at $10 per unit? How many units will be sold?

(B) What happens if the company sets the price at $10 per unit? What will the total revenue be?

(C) Determine the maximum possible revenue and how many units must be sold to achieve the maximum revenue

Answer 1

A). The price of the product (p) and the quantity sold (x) obeys the linear relationship p=100-x/2. Therefore, when p = 10, we have 10 = 100-x/2 = or, x/2 = 100-10 = 90 so that x = 90*2 = 180. Thus, if the company sets the price at $10 per unit, then 180 units will be sold.

B). If the company sets the price at $10 per unit, then the revenue will be price* units sold = $ 10*180 = $ 1800.

C). If x units are sold at the price of $p per unit, the revenue function is R(x) = px = (100-x/2)x = -x²/2+100x = -(1/2)( x²-200x) = -(1/2)( x²-2x*100+100²) +100²/2 = -(1/2)( x-100)²+ 5000.

This is the equation of a downwards opening parabola with vertex at (100, 5000). Since the vertex of a downwards opening parabola is its highest point, hence R(x) will be maximum at $ 5000 when x = 100, i.e. when 100 units are sold.

A graph of R(x) = -x²/2+100x is attached for visual clarity.