Math Problem Statement

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.

Solution

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:

Speed=Change in DistanceChange in Time\text{Speed} = \frac{\text{Change in Distance}}{\text{Change in Time}}

The two time intervals are:

  • From t=0t = 0 to t=8t = 8 (interval [0,8][0, 8]),
  • From t=2t = 2 to t=11t = 11 (interval [2,11][2, 11]).

Interval [0, 8]:

From the graph:

  • At t=0t = 0, the distance is 0 miles.
  • At t=8t = 8, the distance is 1 mile.

The speed in this interval is:

Speed=1080=18=0.125 miles per minute.\text{Speed} = \frac{1 - 0}{8 - 0} = \frac{1}{8} = 0.125 \text{ miles per minute}.

Interval [2, 11]:

From the graph:

  • At t=2t = 2, the distance is 0.4 miles.
  • At t=11t = 11, the distance is 1.5 miles.

The speed in this interval is:

Speed=1.50.4112=1.190.122 miles per minute.\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][0, 8], as her speed was slightly higher: 0.125 miles per minute compared to 0.122 miles per minute during [2,11][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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rate of Change
Distance-Time Graph

Formulas

Speed = Change in Distance / Change in Time

Theorems

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Suitable Grade Level

Grades 6-8

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