engineering
posted by vimal on .
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

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 maximumprofit production level, Q.
3Q^2 +15Q +60 = 0
The positive root is
Q = (1/6)[15 sqrt(289+720)]
Q = 7.8