We are tasked to solve the following problem related to cost functions in economics:

Given:

The marginal cost function (MC) is:
$MC = Q^2 - 42Q + 200$

To determine:

a. The total cost function $C(Q)$ if the fixed cost is $900$.
b. The total cost $C$ when $Q = 12$.

Solution:

Part a: Finding the Total Cost Function $C(Q)$

The marginal cost (MC) is the derivative of the total cost function $C(Q)$: $MC = \frac{dC}{dQ} = Q^2 - 42Q + 200$

To find $C(Q)$, integrate $MC$ with respect to $Q$: $C(Q) = \int MC \, dQ = \int \left( Q^2 - 42Q + 200 \right) dQ$

Performing the integration: $C(Q) = \frac{Q^3}{3} - \frac{42Q^2}{2} + 200Q + K$ where $K$ is the constant of integration, representing the fixed cost.

Given that the fixed cost is $900$, substitute $K = 900$ into the equation: $C(Q) = \frac{Q^3}{3} - 21Q^2 + 200Q + 900$

Thus, the total cost function is: $C(Q) = \frac{Q^3}{3} - 21Q^2 + 200Q + 900$

Part b: Finding Total Cost at $Q = 12$

Substitute $Q = 12$ into the total cost function $C(Q)$: $C(12) = \frac{(12)^3}{3} - 21(12)^2 + 200(12) + 900$

Simplify step-by-step:

• $12^3 = 1728$, so $\frac{12^3}{3} = \frac{1728}{3} = 576$
• $12^2 = 144$, so $21(12)^2 = 21 \cdot 144 = 3024$
• $200(12) = 2400$

Now substitute these values: $C(12) = 576 - 3024 + 2400 + 900$

Combine the terms: $C(12) = (576 + 2400 + 900) - 3024$ $C(12) = 3876 - 3024 = 852$

Final Answers:

a. The total cost function is: $C(Q) = \frac{Q^3}{3} - 21Q^2 + 200Q + 900$

b. The total cost when $Q = 12$ is: $C(12) = 852$

Would you like further clarification or additional steps on this solution?

Here are related questions you might explore:

1. How do we interpret the fixed cost $K$ in economics?
2. What is the role of marginal cost in determining total cost?
3. How do we integrate a polynomial function step-by-step?
4. Can we generalize this method for other cost functions?
5. How is the total cost function used in economic decision-making?

Tip: Always remember that the constant of integration in cost functions represents fixed costs, which do not vary with production levels.