Step 1 :Given the long-run average cost (LRAC) curve: \(\text{Average Costs} = \frac{120}{Q} + 3Q\)
Step 2 :To find the long-run market price, we need to find the minimum point on the LRAC curve. This is done by taking the derivative of the average cost function with respect to Q and setting it equal to zero.
Step 3 :The derivative of the average cost function is: \(\frac{d(\text{Average Costs})}{dQ} = -\frac{120}{Q^2} + 3\)
Step 4 :Setting this equal to zero gives: \(-\frac{120}{Q^2} + 3 = 0\)
Step 5 :Solving for Q gives: \(Q^2 = \frac{-120}{-3}\) which simplifies to \(Q^2 = 40\)
Step 6 :Taking the square root of both sides gives: \(Q = \sqrt{40} = 6.32\) (rounded to two decimal places)
Step 7 :Substituting \(Q = 6.32\) back into the average cost function gives the minimum average cost, which is the long-run market price: \(p^* = \frac{120}{6.32} + 3 * 6.32\)
Step 8 :Solving the above equation gives: \(p^* = 18.99 + 18.96\)
Step 9 :\(p^* = 37.95\) (rounded to two decimal places)
Step 10 :Therefore, the long-run market price must be \(\boxed{37.95}\)