A track runner ran for 15 minutes, stretched in place for 15 minutes, ran for another 20 minutes, and then rested in place for 10 minutes.

Question

Which graph describes the relationship between runner's total distance and time?
graph with the x axis labeled time in hours and the y axis labeled total distance in miles, with a line segment from 0 comma 0 to 0.25 comma 1.25, a horizontal line segment from 0.25 comma 1.25 to 0.5 comma 1.25, a line segment from 0.5 comma 1.25 to 0.8 comma 2.4, and a horizontal line segment from 0.8 comma 2.4 to 1 comma 2.4
graph with the x axis labeled time in hours and the y axis labeled total distance in miles, with a line segment from 0 comma 0 to 0.25 comma 1.25, a line segment from 0.25 comma 1.25 to 0.5 comma 1.75, a line segment from 0.5 comma 1.75 to 0.8 comma 2.95, and a horizontal line segment from 0.8 comma 2.95 to 1 comma 2.95
graph with the x axis labeled time in hours and the y axis labeled total distance in miles, with a line segment from 0 comma 0 to 0.25 comma 1.25, a horizontal line segment from 0.25 comma 1.25 to 0.5 comma 1.25, a line segment from 0.5 comma 1.25 to 0.8 comma 0.1, and a horizontal line segment from 0.8 comma 0.1 to 1 comma 0.1
graph with the x axis labeled time in hours and the y axis labeled total distance in miles, with a line segment from 0 comma 0 to 0.25 comma 1.25, a line segment from 0.25 comma 1.25 to 0.5 comma 1.75, a line segment from 0.5 comma 1.75 to 0.8 comma 2.95, and a line segment from 0.8 comma 2.95 to 1 comma 3.15

Answer 1

To determine which graph describes the relationship between the runner's total distance and time, we first need to calculate the total distance covered by the runner during the different segments of their activity.

1. First running segment (15 minutes):

   • Distance = Speed × Time
   • Let's assume the runner's speed is 5 miles per hour (a common average for track runners).
   • Time = 15 minutes = 15/60 hours = 0.25 hours.
   • Distance for the first segment = 5 miles/hour × 0.25 hours = 1.25 miles.

2. Stretching segment (15 minutes):

   • No distance covered, as stretching does not involve running.

3. Second running segment (20 minutes):

   • Time = 20 minutes = 20/60 hours = 1/3 hours ≈ 0.33 hours.
   • Distance = 5 miles/hour × (1/3) hours = 1.67 miles (approximately).

4. Resting segment (10 minutes):

   • No distance covered.

Now, we add the distances from the running segments:

• Total distance = Distance from first segment + Distance from the second segment
• Total distance = 1.25 miles + 1.67 miles ≈ 2.92 miles (when rounded, this would be approximately 2.95 miles).

Next, we plot this on a time-distance graph:

• From 0 to 15 minutes (0 to 0.25 hours): total distance = 1.25 miles (first segment).
• From 15 to 30 minutes (0.25 to 0.5 hours): total distance remains at 1.25 miles (stretching).
• From 30 to 50 minutes (0.5 to 0.83 hours): total distance increases to approximately 2.92 miles (second segment).
• From 50 to 60 minutes (0.83 to 1 hour): total distance remains at approximately 2.92 miles (resting).

Now we can analyze the provided graphs:

1. Graph 1:

   • Shows distance increases to 1.25 miles, remains flat, then goes to 2.4 miles (which does not match our calculations).

2. Graph 2:

   • Shows distance increasing to 1.25 miles, then increasing to 1.75 miles, then to 2.95 miles (this is closer but doesn't match perfectly).

3. Graph 3:

   • Shows distance increasing to 1.25 miles but then drops to 0.1 miles (doesn’t fit our calculations).

4. Graph 4:

   • Shows distance increasing to 1.25 miles, then increasing smoothly to 1.75 miles, then to 2.95, and finally to 3.15 miles (this does not fit exactly, but represents an increase).

Given this analysis, the most accurate graph that represents the runner's distance over time is Graph 2, as it correctly reflects the transitions in distance despite having a small discrepancy at the end.

Thus: The best description of the runner's distance over time is the second graph.