The function D(t) defines a traveler’s distance from home, in miles, as a function of time, in hours.



Which times and distances are represented by the function? Select three options.

The starting distance, at 0 hours, is 300 miles.
At 2 hours, the traveler is 725 miles from home.
At 2.5 hours, the traveler is still moving farther from home.
At 3 hours, the distance is constant, at 875 miles.
The total distance from home after 6 hours is 1,062.5 miles.

Let's analyze the given statements to see which ones can be supported by the function \( D(t) \), representing the traveler's distance from home at time \( t \):

1. **"The starting distance, at 0 hours, is 300 miles."**
   - This means that \( D(0) = 300 \). This can represent an initial condition, so it's a plausible option.

2. **"At 2 hours, the traveler is 725 miles from home."**
   - This statement gives \( D(2) = 725 \). This is another valid distance at a specific time, so it is a plausible option.

3. **"At 2.5 hours, the traveler is still moving farther from home."**
   - This statement suggests that \( D(t) \) is increasing at \( t = 2.5 \), meaning the traveler is still moving away. This describes a potential trend in the function, and could be plausible if the function is increasing at that point.

4. **"At 3 hours, the distance is constant, at 875 miles."**
   - A constant distance means the traveler has stopped moving at this point, i.e., \( D(t) = 875 \) for \( t \geq 3 \). If this holds for the function, it could be a possible option.

5. **"The total distance from home after 6 hours is 1,062.5 miles."**
   - This gives \( D(6) = 1,062.5 \), a specific value for the distance after 6 hours, which seems reasonable.

### Conclusion:
Based on this, the three most plausible options are:

- The starting distance, at 0 hours, is 300 miles.
- At 2 hours, the traveler is 725 miles from home.
- The total distance from home after 6 hours is 1,062.5 miles.

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### Related questions:
1. What type of function could describe this traveler's movement?
2. How would you calculate the rate of travel between 2 and 3 hours?
3. Could the traveler ever return home based on this distance function?
4. If the traveler maintains a constant speed, how could we estimate it from this data?
5. Can the traveler stop moving while still increasing the distance function?

**Tip:** When analyzing functions, always look for key points like starting values, behavior trends, and maximum/minimum values to better understand the movement or change over time.

The function D(t) defines a traveler’s distance from home, in miles, as a function of time, in hours. Which times and distances are represented by the function? Select three options. The starting distance, at 0 hours, is 300 miles. At 2 hours, the traveler is 725 miles from home. At 2.5 hours, the traveler is still moving farther from home. At 3 hours, the distance is constant, at 875 miles. The total distance from home after 6 hours is 1,062.5 miles.

Learn how to analyze a traveler's distance from home using the function D(t). This problem covers understanding distance as a function of time, interpreting key values at specific time points, and determining trends like constant distance or increasing movement. Suitable for students in Grades 8-10.

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