Final answer:
To maximize profits, a firm in a perfectly competitive market should produce where marginal revenue equals marginal cost. Using the given short-run cost function and market price, the profit-maximizing quantity of output for this firm is 6 units.
Step-by-step explanation:
A firm operating in a perfectly competitive market maximizes profits at the level of output where marginal revenue (MR) equals marginal cost (MC). In this case, the short-run cost function is given as STC(q) = (1/12)q^3 + 3q + 36. To find the profit-maximizing level of output, we need to find the quantity at which MR = MC.
MR is equal to the market price, which is $12 in this case. MC can be calculated by taking the derivative of the total cost function. Differentiating STC(q) with respect to q, we get MC(q) = (1/4)q^2 + 3.
Setting MR equal to MC, we have 12 = (1/4)q^2 + 3. Solving for q, we find that q^2 = 36. Taking the positive square root, we get q = 6. Therefore, the profit-maximizing level of output for this firm is 6 units.