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To determine during which time interval Riley ran the fastest, we need to calculate her speed (rate of change of distance over time) in each interval. The formula for speed is:

\[
\text{Speed} = \frac{\text{Change in Distance}}{\text{Change in Time}}
\]

The two time intervals are:
- From \( t = 0 \) to \( t = 8 \) (interval \([0, 8]\)),
- From \( t = 2 \) to \( t = 11 \) (interval \([2, 11]\)).

### Interval [0, 8]:
From the graph:
- At \( t = 0 \), the distance is 0 miles.
- At \( t = 8 \), the distance is 1 mile.

The speed in this interval is:

\[
\text{Speed} = \frac{1 - 0}{8 - 0} = \frac{1}{8} = 0.125 \text{ miles per minute}.
\]

### Interval [2, 11]:
From the graph:
- At \( t = 2 \), the distance is 0.4 miles.
- At \( t = 11 \), the distance is 1.5 miles.

The speed in this interval is:

\[
\text{Speed} = \frac{1.5 - 0.4}{11 - 2} = \frac{1.1}{9} \approx 0.122 \text{ miles per minute}.
\]

### Conclusion:
Riley ran faster during the time interval \([0, 8]\), as her speed was slightly higher: 0.125 miles per minute compared to 0.122 miles per minute during \([2, 11]\).

Would you like more details or have any questions?

Here are five related questions you can explore:
1. What is the total distance Riley covered during her entire run?
2. What was Riley's average speed over the entire run?
3. Can we model Riley’s motion using a linear equation for each segment?
4. How does acceleration differ from average speed in this context?
5. What factors could affect Riley's running speed over different intervals?

**Tip:** When analyzing graphs, always look for key points that indicate changes in direction, slope, or behavior to better understand motion or other trends.

Riley is training to run a 5-mile race. The graph represents the distance (miles) she ran with respect to time (minutes) in her first training run. Over which time interval did she run the fastest, [0, 8] or [2, 11]? Justify your answer in 1-2 sentences.

Analyze Riley's distance-time graph to determine during which interval she ran the fastest. Using the speed formula, compare the intervals [0, 8] and [2, 11] by calculating the rate of change. Suitable for students in Grades 6-8 learning about distance, time, and speed.

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