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Let's solve the given problem step-by-step.

### Problem Details

We are given:

- **Cost Function**: \( C(x) = 50 + 0.01x^3 \), which models the total cost of producing \( x \) units of a product.
- **Price Function**: \( p(x) = 100 - 0.1x \), which is the price function for \( x \) units.

### Part (a): Find the Marginal Cost Function

The **marginal cost** is the derivative of the cost function \( C(x) \) with respect to \( x \). It represents the cost of producing one additional unit.

\[
\text{Marginal Cost } C'(x) = \frac{d}{dx}(50 + 0.01x^3)
\]

Calculating the derivative:

\[
C'(x) = 0 + 0.01 \cdot 3x^2 = 0.03x^2
\]

So, the marginal cost function is:

\[
\boxed{C'(x) = 0.03x^2}
\]

### Part (b): Set Up an Integral for the Total Revenue Function and Compute the Producer Surplus When 50 Units Are Produced

The **total revenue function** is given by multiplying the price function by the number of units produced:

\[
R(x) = p(x) \cdot x = (100 - 0.1x)x = 100x - 0.1x^2
\]

To find the **producer surplus**, we need to integrate the revenue function up to the production level and subtract the total cost.

Producer Surplus is given by:

\[
\text{Producer Surplus} = \int_{0}^{x} p(q) \, dq - C(x)
\]

For \( x = 50 \), the integral becomes:

\[
\text{Producer Surplus} = \int_{0}^{50} (100 - 0.1q) \, dq - C(50)
\]

1. **Compute the Integral**:

\[
\int (100 - 0.1q) \, dq = 100q - 0.05q^2
\]

Evaluating from 0 to 50:

\[
\left[100q - 0.05q^2 \right]_{0}^{50} = (100 \times 50 - 0.05 \times 50^2) - (100 \times 0 - 0.05 \times 0^2)
\]

\[
= (5000 - 0.05 \times 2500)
= 5000 - 125
= 4875
\]

2. **Calculate the Total Cost \( C(50) \)**:

\[
C(50) = 50 + 0.01 \times 50^3 = 50 + 0.01 \times 125000 = 50 + 1250 = 1300
\]

3. **Compute Producer Surplus**:

\[
\text{Producer Surplus} = 4875 - 1300 = 3575
\]

### Final Answer:

\[
\boxed{\text{Producer Surplus} = 3575}
\]

### Summary

- Marginal Cost Function: \( \boxed{C'(x) = 0.03x^2} \)
- Producer Surplus when 50 units are produced: \( \boxed{3575} \)

Would you like any additional details or explanations on these calculations?

### Related Questions:
1. How do you interpret the marginal cost in economic terms?
2. What is the significance of the producer surplus in economic analysis?
3. How would the producer surplus change if the cost function were different?
4. What is the difference between consumer surplus and producer surplus?
5. How would you find the break-even point using cost and revenue functions?

### Tip:
To better understand economic functions, always visualize the cost, revenue, and price functions on a graph. This helps in understanding the relationship between these variables visually.

An economist uses the cost function C(x) = 50 + 0.01x^3 to model the total cost of producing x units of a product. a) Find the marginal cost function. b) Suppose the price function is p(x) = 100 - 0.1x. Set up an integral for the total revenue function and compute the producer surplus when 50 units are produced.

Learn how to find the marginal cost and producer surplus for a cost function C(x) = 50 + 0.01x^3 and price function p(x) = 100 - 0.1x. This problem involves calculus concepts like integration and differentiation, and is ideal for economics students at the undergraduate level.

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