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Resuelva el siguiente problema, se quiere pasar de 3100 a 3200 toneladas de producción.

The problem presented in the image involves a chemical production company that has the following functions for cost \( C(x) \) and revenue \( R(x) \), where \( x \) represents the number of tons of fertilizer produced per week:

- **Cost function:** \( C(x) = 20,000 + 40x \)
- **Revenue function:** \( R(x) = 100x + 0.01x^2 \)

The company currently produces 3,100 tons per week and is considering increasing production to 3,200 tons per week. The task is to determine the rate of change in profit per ton when increasing production from 3,100 to 3,200 tons.

### Solution:

1. **Profit function:**
   The profit \( P(x) \) is given by the difference between revenue and cost:
   \[
   P(x) = R(x) - C(x)
   \]
   Substituting the given functions:
   \[
   P(x) = (100x + 0.01x^2) - (20,000 + 40x)
   \]
   Simplifying:
   \[
   P(x) = 60x + 0.01x^2 - 20,000
   \]

2. **Rate of change of profit (marginal profit):**
   The marginal profit \( P'(x) \) is the derivative of the profit function with respect to \( x \):
   \[
   P'(x) = \frac{d}{dx} \left( 60x + 0.01x^2 - 20,000 \right)
   \]
   \[
   P'(x) = 60 + 0.02x
   \]

3. **Evaluate at 3,100 and 3,200 tons:**
   We can evaluate the marginal profit at \( x = 3,100 \) and \( x = 3,200 \).

   For \( x = 3,100 \):
   \[
   P'(3100) = 60 + 0.02(3100) = 60 + 62 = 122 \text{ dollars per ton}
   \]

   For \( x = 3,200 \):
   \[
   P'(3200) = 60 + 0.02(3200) = 60 + 64 = 124 \text{ dollars per ton}
   \]

### Conclusion:
The rate of change of profit increases from 122 dollars per ton to 124 dollars per ton when production is increased from 3,100 to 3,200 tons per week.

Would you like more details on any part of this solution, or do you have any further questions? Here are some related questions to explore:

1. How do you calculate the marginal cost from the cost function?
2. What is the significance of the marginal profit in economic terms?
3. How would the profit function change if the cost function were quadratic instead of linear?
4. What does the second derivative of the profit function tell us?
5. How do changes in fixed costs affect the overall profit function?

**Tip:** When analyzing cost and revenue functions, always check if the functions include non-linear terms, as these can significantly impact the marginal calculations.

Explore the calculation of marginal profit when a chemical production company increases its weekly production from 3100 to 3200 tons. Understand the profit function, revenue and cost equations involved, and the rate of change in profit per ton. Detailed step-by-step solution provided.

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