Question:\(\mathrm{R(x) = \frac{40x + 160}{x + 2}}\). The function above models the daily revenue R, in hundreds of dollars, for...

1. TRANSLATE the problem information

• Given: \(\mathrm{R(x) = \frac{40x + 160}{x + 2}}\) models daily revenue in hundreds of dollars
• Find: What does the number 40 represent in this context?
• The answer choices suggest we need to understand different aspects of the function's behavior

2. INFER the approach needed

• Since we're asked what a specific number "represents," we need to analyze how that number appears in the function's behavior
• The number 40 appears in the numerator, but its true meaning comes from the function's long-term behavior
• We should look at what happens to R(x) as time increases (\(\mathrm{x \to \infty}\))

3. SIMPLIFY the rational function to reveal its structure

• Rewrite the numerator: \(\mathrm{40x + 160 = 40(x + 2) + 80}\)
• This gives us: \(\mathrm{R(x) = \frac{40(x + 2) + 80}{x + 2}}\)
• Split the fraction: \(\mathrm{R(x) = \frac{40(x + 2)}{x + 2} + \frac{80}{x + 2}}\)
• Simplify: \(\mathrm{R(x) = 40 + \frac{80}{x + 2}}\)

4. INFER the long-term behavior

• As x increases (more months pass), the term \(\mathrm{\frac{80}{x + 2}}\) gets smaller and smaller
• Eventually: \(\mathrm{\lim_{x\to\infty} R(x) = 40 + 0 = 40}\)
• This means the daily revenue approaches 40 hundreds of dollars over time

5. APPLY CONSTRAINTS to select the correct interpretation

• Check what 40 represents: the horizontal asymptote value
• This matches choice (D): "The daily revenue that the business approaches as time increases"

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students see "40" in the function and immediately calculate R(0) to find the initial revenue, thinking that's what the coefficient represents.

When \(\mathrm{x = 0}\):

\(\mathrm{R(0) = \frac{0 + 160}{0 + 2} = 80}\) hundreds of dollars

This gives them 80 as the initial revenue, and they might incorrectly think 40 is half of that or represents some other relationship. Since 40 doesn't match the initial revenue, they get confused and may guess or select Choice (A) (initial revenue) thinking there's an error in their calculation.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students recognize they need to analyze the function's behavior but don't properly rewrite the rational function to reveal its asymptotic structure.

Without seeing \(\mathrm{R(x) = 40 + \frac{80}{x + 2}}\), they can't identify that 40 is the horizontal asymptote. They might try to find maximums, derivatives, or other function properties, leading to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students understand that coefficients in rational functions often reveal themselves through asymptotic behavior, not just direct substitution. The key insight is recognizing that "what does 40 represent" requires analyzing long-term function behavior, not initial conditions.