Final answer:
The marginal cost and average cost at Q = 50 are derived from the total cost function. For Q = 50, the marginal cost is 3007, and the average cost is approximately 1507.24.
Step-by-step explanation:
The question pertains to finding the marginal cost and average cost for a given total cost function, Tc = 30Q2 + 7Q + 12, where Q represents the quantity of goods produced. To find these costs when Q = 50, we first calculate the marginal cost by taking the derivative of the total cost function with respect to Q, and then we evaluate it at Q = 50. The marginal cost (MC) formula is MC = d(Tc)/dQ. To find the average cost, we divide the total cost by the quantity, which yields the average cost (AC) formula as AC = Tc/Q.
Calculating the marginal cost:
- First, take the derivative of the total cost function: MC = d(30Q2 + 7Q + 12)/dQ = 60Q + 7.
- Then, evaluate it at Q = 50: MC = 60(50) + 7 = 3007.
Calculating the average cost:
- Divide the total cost by the quantity at Q = 50: AC = (30(50)2 + 7(50) + 12) / 50.
- Calculate the result: AC = (75,000 + 350 + 12) / 50 = 1507.24.
Therefore, the marginal cost at Q = 50 is 3007, and the average cost is approximately 1507.24.