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The problem presents a revenue function \( R(n) \) where \( n \) represents hundreds of units sold, and \( R(n) \) gives the revenue in thousands of dollars.

The function is:

\[
R(n) = 20 \frac{(n - 24)}{(n - 25)^2 + n - 24}
\]

The goal is to determine for which values of \( n \) the revenue is negative, i.e., when \( R(n) < 0 \).

### Step 1: Analyze the equation for \( R(n) \)
We need to find the range of values of \( n \) where:

\[
R(n) = 20 \frac{(n - 24)}{(n - 25)^2 + n - 24} < 0
\]

This inequality will hold when the numerator and the denominator have opposite signs. Specifically, the sign of the function depends on the numerator \( n - 24 \), as the denominator is always positive (since it's the sum of squares and positive terms).

- **Numerator**: \( n - 24 \)
  - \( n - 24 < 0 \) when \( n < 24 \)
  - \( n - 24 > 0 \) when \( n > 24 \)

- **Denominator**: \( (n - 25)^2 + n - 24 \)
  - This expression is always positive for all values of \( n \), because \( (n - 25)^2 \geq 0 \) and \( n - 24 \geq -24 \).

Thus, \( R(n) \) will be negative when \( n - 24 < 0 \), i.e., when \( n < 24 \).

### Step 2: Conclusion
The range of values of \( n \) that result in negative revenue is \( n < 24 \).

The correct answer is:
\[
\boxed{B. (-\infty, 24)}
\]

Let me know if you'd like a further breakdown of any steps!

### Here are 5 related questions:
1. How would the graph of \( R(n) \) look for different values of \( n \)?
2. What happens to \( R(n) \) as \( n \) approaches 24 from the left and from the right?
3. How does the denominator influence the behavior of \( R(n) \) near \( n = 25 \)?
4. What is the significance of the value \( n = 24 \) in this revenue model?
5. How would you find the maximum revenue generated according to this model?

**Tip:** When solving inequalities involving rational functions, always check the signs of both the numerator and the denominator separately!

Para maximizar sus ganancias, una compañía contrató a un especialista para determinar un modelo que pudiera representar los ingresos generados al vender una cierta cantidad de su producto. El especialista determinó la siguiente función que modela los ingresos: R(n) = 20 * (n - 24) / ((n - 25)^2 + n - 24). Donde R son los ingresos en miles de dólares y n representa cientos de unidades vendidas. Asumiendo que el modelo es correcto, ¿Cuál es el rango de valores de 'n' que resultarán en ganancias negativas?

Learn how to determine the range of values for which the revenue model R(n) = 20 * (n - 24) / ((n - 25)^2 + n - 24) results in negative revenue. This problem covers rational functions, inequality solving, and revenue analysis in thousands of dollars for a product's sales. Suitable for students in Grades 10-12.

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