Final answer:
The marginal cost function, derived from the variable cost function VC = 120Q - 9Q² + 0.25Q³, is MC(Q) = 120 - 18Q + 0.75Q². Marginal cost reflects the additional cost of producing one more unit and does not include fixed costs like the $180 mentioned.
Step-by-step explanation:
To determine the marginal cost function, which is the cost of producing one additional unit of output, we need to take the derivative of the variable cost function. The given variable cost (VC) function is VC = 120Q - 9Q² + 0.25Q³. The marginal cost (MC) is the derivative of this function with respect to Q, the quantity of output.
The derivative of each term in the VC function:
- 120Q becomes 120,
- -9Q² becomes -18Q,
- 0.25Q³ becomes 0.75Q².
So, the marginal cost function, MC(Q), is equal to 120 - 18Q + 0.75Q².
It's important to distinguish that fixed costs, such as the $180 given in the problem, do not affect the marginal cost because they do not change with the level of production. Therefore, fixed costs are not considered when calculating the marginal cost, which focuses on variable costs only. As production increases, the variable costs added to the fixed costs lead to a total cost that includes both these components.