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the graph shows dots on (0,0), (2,0.4), (8,1), (11,1.5)

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?

(1 point)
Responses

[0, 8]
[0, 8]

[2, 11]

3 answers

To determine the time interval over which Riley ran the fastest, we need to find the slope of the line segments between the points on the graph, as the slope represents the speed (distance per unit time) during those intervals.

  1. From (0,0) to (2,0.4):

    • Change in distance: 0.4 - 0 = 0.4 miles
    • Change in time: 2 - 0 = 2 minutes
    • Slope = 0.4 miles / 2 minutes = 0.2 miles/min.

  2. From (2,0.4) to (8,1):

    • Change in distance: 1 - 0.4 = 0.6 miles
    • Change in time: 8 - 2 = 6 minutes
    • Slope = 0.6 miles / 6 minutes = 0.1 miles/min.

  3. From (8,1) to (11,1.5):

    • Change in distance: 1.5 - 1 = 0.5 miles
    • Change in time: 11 - 8 = 3 minutes
    • Slope = 0.5 miles / 3 minutes = approximately 0.17 miles/min.

Now we can compare the slopes:

  • The fastest speed is during the interval [0, 2].

Therefore, Riley ran the fastest during the time interval [0, 2].

the answer can only be (0, 8) or (2,11)

Given the options [0, 8] and [2, 11], let’s analyze the intervals based on the slopes calculated earlier:

  1. Interval [0, 8] includes:

    • From (0,0) to (2,0.4):
      • Slope = 0.2 miles/min
    • From (2,0.4) to (8,1):
      • Slope = 0.1 miles/min
    • Composite slope for the interval [0, 8]:
      • Total distance = 1 mile (from 0 to 1)
      • Total time = 8 minutes
      • Slope = 1 mile / 8 minutes = 0.125 miles/min.

  2. Interval [2, 11] includes:

    • From (2,0.4) to (8,1):
      • Slope = 0.1 miles/min
    • From (8,1) to (11,1.5):
      • Slope ≈ 0.17 miles/min
    • Composite slope for the interval [2, 11]:
      • Total distance = 1.1 miles (from 0.4 to 1.5)
      • Total time = 9 minutes
      • Slope = 1.1 miles / 9 minutes ≈ 0.122 miles/min.

Based on these calculations, the fastest running speed occurs in the interval [0, 8], since this interval contains the highest slope (0.2 miles/min) of the segments included.

Therefore, the answer is [0, 8].