Q40004p where P is selling price per unit and Q is quanitity

Q=4000-4p, where P is selling price per unit and Q is quanitity demanded.   V= $25/unit which is the revenue per unit and Fixed cost = $10025. (a) Find Breakeven quanitity or quanitties, (b) find maximum revenue, (c) fidn the maximum profit.

Solution

Q = 4000 - 4P

P = (4000 - Q) / 4 = 1000 - 0.25Q

(a)

If breakeven quantity be Q, then

Revenue = Variable cost + Fixed cost

P x Q = 25Q + 10025

Q x (1000 - 0.25Q) = 25Q + 10025

1000Q - 0.25Q² = 25Q + 10025

0.25Q² - 975Q + 10025 = 0

Dividing by 0.25,

Q² - 3900Q + 40100 = 0

This is a quadratic equation, solving which** we get:

Q = 3890 or Q = 10

**Because of large coefficients, this equation has been solved using online quadratic equation solver tool.

(b)

Revenue, R = P x Q = 1000Q - 0.25Q² [From part (a)]

Revenue is maximum when dR / dQ = 0

1000 - 0.5Q = 0

0.5Q = 1000

Q = 2000

P = 1000 - 0.25Q = 1000 - 500 = 500

Maximum revenue = P x Q = 500 x 2000 = 1,000,000

(c)

Profit, Z = Revenue - VC - FC

Z = 1000Q - 0.25Q² - 25Q - 10025

Z = 975Q - 0.25Q² - 10025

Profit is maximum when dZ / dQ = 0

975 - 0.5Q = 0

0.5Q = 975

Q = 1950

Maximum Z = 975Q - 0.25Q² - 10025 = (975 x 1950) - (0.25 x 1950 x 1950) - 10025

= 1,901,250 - 950,625 - 10025

= 940,600