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Let's go through each part of the problem step by step:

### Given information:
- The marginal cost function is $ C'(x) = -0.04x + 22 $, where $ x $ represents the level of production and $ C'(x) $ represents the marginal cost.

#### Part a: Compute $ C'(510) $

We are asked to compute the marginal cost at a production level of 510 items:

\[
C'(510) = -0.04(510) + 22
\]

Let's calculate:

\[
C'(510) = -20.4 + 22 = 1.6
\]

So, the marginal cost at 510 items is **$1.60** per item.

#### Part b: Integrate $ C'(x) $ to find the cost function $ C(x) $

To find $ C(x) $, we need to integrate the marginal cost function $ C'(x) = -0.04x + 22 $:

\[
C(x) = \int (-0.04x + 22) \, dx
\]

The integration gives:

\[
C(x) = -0.02x^2 + 22x + C_0
\]

Here, $ C_0 $ is the constant of integration. To find $ C_0 $, we use the given information that the cost at a production level of 100 items is known to be $10,000. So, we substitute $ x = 100 $ and $ C(100) = 10,000 $:

\[
10,000 = -0.02(100)^2 + 22(100) + C_0
\]
\[
10,000 = -0.02(10,000) + 2,200 + C_0
\]
\[
10,000 = -200 + 2,200 + C_0
\]
\[
C_0 = 10,000 - 2,000 = 8,000
\]

So the cost function is:

\[
C(x) = -0.02x^2 + 22x + 8,000
\]

#### Part c: Compute $ C(510) $

Now we are asked to compute the total cost at a production level of 510 items using the cost function we just derived:

\[
C(510) = -0.02(510)^2 + 22(510) + 8,000
\]

Let's calculate:

\[
C(510) = -0.02(260,100) + 22(510) + 8,000
\]
\[
C(510) = -5,202 + 11,220 + 8,000
\]
\[
C(510) = 14,018
\]

So, the total cost at a production level of 510 items is **$14,018**.

### Summary of answers:
- **Part a:** $ C'(510) = 1.60 $ per item
- **Part b:** $ C(x) = -0.02x^2 + 22x + 8,000 $
- **Part c:** $ C(510) = 14,018 $

Let me know if you'd like more details or have any questions!

Here are five questions that relate to this problem:
1. What is the interpretation of the marginal cost at different production levels?
2. How does the constant $ C_0 $ affect the overall cost function?
3. What would happen to the cost if the marginal cost function had a different slope?
4. How would you compute the break-even point if a price function were given?
5. How does integrating a marginal cost function relate to total cost?

**Tip:** Always double-check units and ensure rounding is consistent with the instructions given (e.g., to the nearest cent or dollar).

Let's consider the marginal cost function C'(x) = -0.04x + 22, where x represents the level of production and C'(x) represents the marginal cost. At the production level of 100 items, the cost is known to be $10,000. Compute C'(510), integrate C'(x) to find C(x), and compute C(510).

Learn how to compute the marginal cost at a production level of 510 items, integrate the marginal cost function, and calculate the total cost for 510 items. This problem covers key calculus concepts such as marginal cost, integration, and quadratic functions. Suitable for students in Grades 11-12.

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