To solve the problem, let's find the total cost to produce 40 items, $C(40)$, based on the provided fixed cost and marginal cost function:

Given:

1. Fixed cost = $1800.
2. Marginal cost function: $C'(q) = q^2 + 10q - 90$.

Approach:

The total cost $C(q)$ is obtained by integrating the marginal cost function $C'(q)$ and then adding the fixed cost.

Step 1: Integrate the marginal cost function

$C(q) = \int (q^2 + 10q - 90) \, dq + \text{Fixed Cost}.$ Perform the integration: $\int (q^2 + 10q - 90) \, dq = \frac{q^3}{3} + 5q^2 - 90q + C_0,$ where $C_0$ is the constant of integration.

Step 2: Apply the fixed cost

The fixed cost of $1800 is the initial total cost when no items are produced ($q = 0$), so: $C(q) = \frac{q^3}{3} + 5q^2 - 90q + 1800.$

Step 3: Evaluate $C(40)$

Substitute $q = 40$ into the total cost function: $C(40) = \frac{40^3}{3} + 5(40^2) - 90(40) + 1800.$

Step 4: Simplify the terms

• $\frac{40^3}{3} = \frac{64000}{3} \approx 21333.33$,
• $5(40^2) = 5(1600) = 8000$,
• $-90(40) = -3600$.

Now, sum these values: $C(40) \approx 21333.33 + 8000 - 3600 + 1800 = 27533.33.$

Thus, the total cost to produce 40 items is approximately $27533.

Final Answer:

$\boxed{27533}$

Do you have any further questions or want a deeper explanation?

Related Questions:

1. How is the fixed cost used in calculating the total cost function?
2. What does the marginal cost function represent in practical terms?
3. Why do we integrate the marginal cost to find the total cost?
4. What happens if the fixed cost is increased? How would the result change?
5. How can the cost function be used to find the average cost per item?

Tip: Always verify the units (e.g., dollars, items) when solving cost problems to ensure consistency.