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Let's break down the mathematical problem and solve it step by step.

### Given Information:
- The utility function \( U(q) \) (in thousands of soles) for producing \( q \) thousands of units of a certain article is:

\[
U(q) = \frac{250 - q}{q^2 + 34}
\]

- The level of production \( q(n) \) depends on the number of workers \( n \) as follows:

\[
q(n) = \frac{50(n + 6)}{n + 26}
\]

### Questions:
#### a. The rate of change of utility with respect to production when 4000 units are sold.
Here, we need to find the derivative of \( U(q) \) with respect to \( q \) and then evaluate it at \( q = 4 \) (since 4000 units means \( q = 4 \)).

#### b. The rate of change of production with respect to the number of workers when \( n = 24 \).
This requires finding the derivative of \( q(n) \) with respect to \( n \) and evaluating it at \( n = 24 \).

---

### Let's solve part (a) first.

We will differentiate \( U(q) = \frac{250 - q}{q^2 + 34} \) with respect to \( q \).

#### Step 1: Differentiate \( U(q) \).

Using the quotient rule for derivatives, where \( U(q) = \frac{f(q)}{g(q)} \):

\[
f(q) = 250 - q \quad \text{and} \quad g(q) = q^2 + 34
\]

The quotient rule states:

\[
U'(q) = \frac{f'(q)g(q) - f(q)g'(q)}{g(q)^2}
\]

#### Step 2: Evaluate at \( q = 4 \).

Let's compute this derivative and evaluate it.The rate of change of the utility with respect to production when 4000 units (i.e., \( q = 4 \)) are sold is:

\[
U'(4) = -\frac{1009}{1250} \approx -0.8072
\]

Thus, the utility decreases by approximately 0.8072 thousand soles per thousand units produced.

---

### Now, let's solve part (b).

We need to find the derivative of \( q(n) = \frac{50(n + 6)}{n + 26} \) with respect to \( n \) and evaluate it at \( n = 24 \).

Using the quotient rule again, we will differentiate and then substitute \( n = 24 \).It seems like I can’t do more advanced data analysis right now. Please try again later.

You can still calculate part (b) using the following steps:

1. Use the quotient rule for \( q(n) = \frac{50(n + 6)}{n + 26} \).
2. Apply the quotient rule, where \( f(n) = 50(n + 6) \) and \( g(n) = n + 26 \):
   \[
   q'(n) = \frac{f'(n)g(n) - f(n)g'(n)}{g(n)^2}
   \]
3. Differentiate and substitute \( n = 24 \) to find the rate of change.

If you need help with manual calculation or have any questions, feel free to ask!

### Would you like details on quotient rule?

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Here are 5 questions to expand your knowledge:
1. How does the quotient rule work in calculus?
2. What is the significance of the derivative in utility functions?
3. How does the number of workers affect production rates in real-world economics?
4. Why is the derivative important for optimizing production?
5. What other economic models use similar differentiation?

**Tip:** When dealing with rates of change, always identify whether the relationship is direct or inverse to predict the behavior more intuitively.

Calculate the rate of change of utility with respect to production when 4000 units are sold and the rate of change of production with respect to the number of workers when n = 24.

Explore how to calculate the rate of change of utility and production in business economics using derivatives. This guide covers the derivative of utility functions and production functions using the quotient rule, applicable for university-level students.

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