High School

A cab company offers a special discount on fares to senior citizens. The following expression models the average amount a cab driver of the company collects on a particular day, where [tex]$x$[/tex] represents the number of senior citizens who travel by the company's cabs:

[tex]\[
\frac{180x}{x+4} + 250
\][/tex]

What does the constant term in the above expression represent?

A. The constant 250 represents the maximum amount a cab driver collects on a particular day when [tex]$x$[/tex] senior citizens travel by the company's cabs.

B. The constant 250 represents the number of senior citizens who travel by the company's cabs.

C. The constant 250 represents the average amount a cab driver collects on a particular day when no senior citizens travel by the company's cabs.

D. The constant 250 represents the average amount the company pays a cab driver on a particular day.

Answer :

To solve this question, let's analyze the given expression:

[tex]\[
\frac{180x}{x+4} + 250
\][/tex]

Here, [tex]\(x\)[/tex] represents the number of senior citizens who travel by the company's cabs. We need to identify what the constant term [tex]\(250\)[/tex] represents.

Step-by-Step Explanation:

1. Understand the Expression:
The expression [tex]\(\frac{180x}{x+4} + 250\)[/tex] models the average amount a cab driver collects. The term [tex]\(\frac{180x}{x+4}\)[/tex] depends on the number of senior citizens traveling. The value of this term changes with different values of [tex]\(x\)[/tex].

2. Role of the Constant Term:
The constant term in the expression is [tex]\(250\)[/tex], and it doesn't depend on [tex]\(x\)[/tex]. It remains unchanged regardless of how many senior citizens ([tex]\(x\)[/tex]) are traveling.

3. Interpreting the Constant Term (250):
- When no senior citizens travel, set [tex]\(x = 0\)[/tex]. Substitute [tex]\(x = 0\)[/tex] into the expression:
[tex]\[
\frac{180 \times 0}{0 + 4} + 250 = \frac{0}{4} + 250 = 0 + 250 = 250
\][/tex]
- This shows that when no senior citizens are involved, the average amount the cab driver collects is [tex]\(250\)[/tex].

4. Conclusion:
- The constant term [tex]\(250\)[/tex] specifically represents the average amount a cab driver collects on a day when no senior citizens travel by the company's cabs.

Therefore, the correct answer is C: The constant 250 represents the average amount a cab driver collects on a particular day when no senior citizens travel by the company's cabs.