2. 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

To calculate the level of output Q that maximizes total revenue, we need to find the quantity at which the marginal revenue is zero. To do this, we need to differentiate the demand function with respect to quantity, Q, and set it equal to zero.

a) Maximizing total revenue:
Step 1: Calculate the marginal revenue (MR) by differentiating the demand function, P, with respect to Q.
Given demand function: P = 24 - 0.5Q
Differentiate P with respect to Q:
dP/dQ = -0.5
MR = -0.5Q

Step 2: Set marginal revenue (MR) equal to zero and solve for Q.
-0.5Q = 0
Q = 0

Therefore, the level of output Q that maximizes total revenue is Q = 0.

b) Maximizing profits:
To maximize profits, we need to find the level of output where the difference between total revenue and total cost is maximized. This occurs at the point where marginal revenue (MR) equals marginal cost (MC).

Step 1: Calculate the marginal cost (MC) by differentiating the average cost function, AC, with respect to quantity Q.
Given average cost function: AC = Q^2 - 8Q + 36 + 3/Q
Differentiate AC with respect to Q:
dAC/dQ = 2Q - 8 - 3/Q^2
MC = 2Q - 8 - 3/Q^2

Step 2: Set marginal revenue (MR) equal to marginal cost (MC) and solve for Q.
-0.5Q = 2Q - 8 - 3/Q^2
Multiply through by Q^2 to eliminate the fraction:
-0.5Q^3 = 2Q^3 - 8Q^2 - 3
Combine like terms:
2.5Q^3 - 8Q^2 + 3 = 0

Unfortunately, finding the exact solution to this cubic equation requires more advanced mathematical techniques than can be accomplished here. You may wish to use numerical approximation methods (such as Newton's method) or software like Excel or Mathematica to find the value of Q that maximizes profits.

Alternatively, you can graph the total revenue and total cost functions, and identify the level of output where the gap between the two curves is greatest. This corresponds to the level of output that maximizes profits.