Question

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. The points are as follows: (0, 0), (2, 0.4), (8, 1), (11, 1.5)
a. [0, 8]
b. [2, 11]
c. [2, 8]

Answer 1

To determine over which time interval Riley ran the fastest, we need to find the slope (rate of distance per time) between the given points. The slope is calculated as the change in distance divided by the change in time.

Let's calculate the speed over each of the suggested intervals:

1. **Interval [0, 8]:**
- Points: (0, 0) and (8, 1)
- Distance: 1 mile - 0 miles = 1 mile
- Time: 8 minutes - 0 minutes = 8 minutes
- Speed = \( \frac{1 \text{ mile}}{8 \text{ minutes}} = 0.125 \text{ miles/minute} \)

2. **Interval [2, 11]:**
- Points: (2, 0.4) and (11, 1.5)
- Distance: 1.5 miles - 0.4 miles = 1.1 miles
- Time: 11 minutes - 2 minutes = 9 minutes
- Speed = \( \frac{1.1 \text{ miles}}{9 \text{ minutes}} \approx 0.122 \text{ miles/minute} \)

3. **Interval [2, 8]:**
- Points: (2, 0.4) and (8, 1)
- Distance: 1 mile - 0.4 miles = 0.6 miles
- Time: 8 minutes - 2 minutes = 6 minutes
- Speed = \( \frac{0.6 \text{ miles}}{6 \text{ minutes}} = 0.1 \text{ miles/minute} \)

Now, let's summarize the speeds for the time intervals:
- Speed over [0, 8]: 0.125 miles/minute
- Speed over [2, 11]: 0.122 miles/minute
- Speed over [2, 8]: 0.1 miles/minute

From these calculations, we can see that Riley ran the fastest over the interval **[0, 8]**, as it has the highest speed of 0.125 miles/minute.

Thus, the answer is:
**a. [0, 8]**