To solve this problem, let's analyze the information provided and identify the correct function $f(x)$ that best models the total distance traveled by the bus $x$ hours after it starts.

Analyzing the Data:

The table shows:

• At $x = 0$ hours, the total distance is 10 miles.
• At $x = 5$ hours, the distance is 85 miles.
• At $x = 10$ hours, the distance is 160 miles.
• At $x = 15$ hours, the distance is 235 miles.

We can observe that the total distance increases by a constant amount over equal intervals of time, suggesting a linear relationship between $x$ and the distance traveled.

Finding the Rate of Change:

The increase in distance over each 5-hour interval is: $85 - 10 = 75 \quad \text{(from 0 to 5 hours)}$ $160 - 85 = 75 \quad \text{(from 5 to 10 hours)}$ $235 - 160 = 75 \quad \text{(from 10 to 15 hours)}$

So, the bus travels an additional 75 miles every 5 hours, which gives a rate of: $\frac{75}{5} = 15 \text{ miles per hour}$

Deriving the Linear Function:

The distance traveled after $x$ hours can be represented by a linear function: $f(x) = \text{initial distance} + (\text{rate of change}) \cdot x$

Since the initial distance (when $x = 0$) is 10 miles, and the rate of change is 15 miles per hour, the function becomes: $f(x) = 10 + 15x$

Choosing the Correct Answer:

The answer that matches this function is: $\boxed{f(x) = 10 + 15x}$

Final Answer:

The correct option is:

• $f(x) = 10 + 15x$

Would you like further explanation on any steps?

Here are 5 related questions to explore further:

1. How can you determine if a data set represents a linear relationship?
2. What is the significance of the slope in a linear function?
3. How would the function change if the bus initially started at a different distance?
4. What is the importance of the initial value in real-world applications of linear functions?
5. How would this problem differ if the bus's rate of travel changed over time?

Tip:

In problems involving linear relationships, look for a constant rate of change in intervals to confirm linearity.