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To determine which function represents the situation, we need to analyze the data in the table to find the correct equation that models the relationship between \( n \) (number of students) and \( r(n) \) (total cost in dollars).

The table provides values for \( n \) and \( r(n) \):
- When \( n = 0 \), \( r(n) = 55 \)
- When \( n = 5 \), \( r(n) = 130 \)
- When \( n = 15 \), \( r(n) = 280 \)
- When \( n = 30 \), \( r(n) = 505 \)
- When \( n = 35 \), \( r(n) = 580 \)

### Step 1: Find the rate of change
We can calculate the rate of change (slope) by finding the difference in \( r(n) \) values for consistent differences in \( n \). For instance:
- From \( n = 5 \) to \( n = 15 \), \( r(n) \) changes from 130 to 280. This is a difference of:
  \[
  280 - 130 = 150
  \]
  Over a change of \( n = 15 - 5 = 10 \), the rate of change is:
  \[
  \frac{150}{10} = 15
  \]

This rate of change is consistent in other intervals as well, confirming that the function has a slope of 15.

### Step 2: Determine the y-intercept
Since the rate of change is 15, our function has the form:
\[
r(n) = 15n + b
\]
where \( b \) is the y-intercept.

When \( n = 0 \), \( r(n) = 55 \), so:
\[
55 = 15 \cdot 0 + b
\]
\[
b = 55
\]

### Step 3: Write the equation
With a slope of 15 and a y-intercept of 55, the equation is:
\[
r(n) = 15n + 55
\]

### Conclusion
The correct answer is:
**D. \( r(n) = 15n + 55 \)**

Would you like more details on this solution or have any questions?

Here are some related questions:
1. How can you determine the slope of a linear function from a table?
2. What is the significance of the y-intercept in a function like this?
3. How would you identify a non-linear relationship from a table of values?
4. Why is it useful to check multiple intervals to confirm the slope?
5. How can you use a table of values to validate a given equation?

**Tip:** Checking more than one interval in a table ensures accuracy when identifying the rate of change, especially in linear functions.

The table below shows function r, which represents the total cost of a field trip to the science museum, in dollars, if n students return permission slips to attend.

Which function represents the situation?

A. r(n) = 5n + 45
B. r(n) = 30n - 20
C. r(n) = 55n + 15
D. r(n) = 15n + 55

Learn how to determine the linear function that represents the total cost of a field trip to the science museum based on the number of students attending. This problem involves analyzing slope, intercept, and formulating a linear equation suitable for Grades 7-9.

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