Final answer:
To maximize profit at a market price of $180, a perfectly competitive firm should produce 30 units of output, equating the given market price with the marginal cost.
Step-by-step explanation:
Maximizing Profit for a Perfectly Competitive Firm
For a perfectly competitive firm to maximize its profit, it must produce output at the level where the market price is equal to the marginal cost (MC) of production. Given that the market price of the product is $180, and the marginal cost (MC) is given by 6q, where q is the quantity of output, the firm should set MC equal to the price to find the profit-maximizing quantity. This equates to 6q = 180, simplifying to q = 30 units of output. The fixed cost (F) of $3000 is a sunk cost in the short run and does not affect the decision of how much to produce.
Thus, to maximize profits, it is necessary to consider the total costs, which are the sum of fixed costs and variable costs (VC). Since VC = 3q2 and F = $3000, the total cost (TC) equation becomes TC = 3q2 + 3000. However, the optimal output quantity has already been determined by equating MC with the price.
In conclusion, the firm should produce 30 units of output to maximize profit, given the market price of $180. This is the output level where the firm's marginal revenue, which is equal to the market price, aligns with the marginal cost.