Suppose that the market is made up of many identical firms. Each firm's Long-Run Average Cost curve is given by:

Average Costs $=((120 / Q)+3 Q)$

And their Marginal Cost curve is given by:
\[
M C=2 * 3 * Q
\]

Given this, what must be the Long-Run market price $\left(\mathrm{p}^{*}\right)$ if there exist free entry and exit of firms?
Note: round your answers to two decimal places.

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 \(\boxed{37.95}\)