Posted by **vimal** on Friday, October 8, 2010 at 5:29am.

Given a firm’s demand function, P = 24 - 0.5Q and the average cost function, AC = Q2 – 8Q + 36 + 3/Q, calculate the level of output Q which a) maximizes total revenue b) maximizes profits

- engineering -
**drwls**, Friday, October 8, 2010 at 6:06am
Revenue = Quantity*Price

R = Q*P = 24Q - .5Q^2

Revenue is a maximum when

dR/dQ = 0

24 -Q = 0

Q = 24 units sold @ a price of P = 12

Profit (P) is

P = R - Q*(AC)= Q*P - Q*(AC)

= 24Q -0.5Q^2 -Q*(Q^2 -8Q +36 +3/Q)

= 24Q -0.5Q^2 -Q^3 +8Q^2 +36Q +3

= -Q^3 +7.5 Q^2 +60Q +3

Set dP/dQ = 0 to solve for the maximum-profit production level, Q.

-3Q^2 +15Q +60 = 0

The positive root is

Q = (-1/6)[-15 -sqrt(289+720)]

Q = 7.8

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