To solve this problem, let's break it down and consider two different interpretations of "distance" as mentioned: linear distance (as if measured on a straight line) and arc distance (distance along the circular track).

Given Data:

• Diameter of the track = 80 meters.
• Radius of the track, $r = \frac{80}{2} = 40$ meters.
• Circumference of the circular track = $C = 2\pi r = 2\pi \times 40 = 80\pi \approx 251.2$ meters.
• Hillary's speed = 300 meters per minute.
• Eugene's speed = 240 meters per minute.
• Time spent running = 50 minutes.

Step 1: Find the Total Distance Each Person Runs

• Hillary's total distance = $300 \, \text{m/min} \times 50 \, \text{min} = 15,000 \, \text{m}$.
• Eugene's total distance = $240 \, \text{m/min} \times 50 \, \text{min} = 12,000 \, \text{m}$.

Step 2: Calculate the Number of Laps Completed by Each Runner

The number of laps around the track is the total distance divided by the circumference of the track:

• Hillary's laps = $\frac{15,000}{80\pi} \approx 59.7 \, \text{laps}$.
• Eugene's laps = $\frac{12,000}{80\pi} \approx 47.8 \, \text{laps}$.

Step 3: Calculate Their Final Positions on the Track

Since only the fractional part of the laps matters for the final position:

• Hillary's final position after 59.7 laps is $0.7 \times 80\pi \approx 175.84 \, \text{m}$ from the starting point.
• Eugene's final position after 47.8 laps is $0.8 \times 80\pi \approx 201 \, \text{m}$ from the starting point.

Step 4: Calculate the Arc Distance Between Them

Since they are running in opposite directions, the arc distance is found by calculating the total arc length along the circle:

• The total circumference is $80\pi \approx 251.2$ meters.
• Their arc distance = Circumference - |Hillary’s final position - Eugene’s final position| $\text{Arc distance} = 251.2 - |201 - 175.84| = 251.2 - 25.16 = 226.04 \, \text{m}.$

Step 5: Calculate the Linear Distance Between Them

For linear distance, we treat the track as a circle and find the straight-line distance (chord) between their positions:

• The angle between their positions, $\theta$, is proportional to the arc length divided by the total circumference. $\theta = \frac{226.04}{251.2} \times 360^\circ \approx 323.4^\circ.$ The chord length is given by: $\text{Linear distance} = 2r \sin\left(\frac{\theta}{2}\right) = 2 \times 40 \times \sin\left(\frac{323.4^\circ}{2}\right) \approx 2 \times 40 \times \sin(161.7^\circ) \approx 79.85 \, \text{m}.$

Summary of Results:

• Arc Distance between Hillary and Eugene: Approximately 226.04 meters.
• Linear Distance (chord) between them: Approximately 79.85 meters.

These are the two interpretations of "distance" on a circular track.

Would you like further clarification on any part of this solution?

5 Related Questions:

1. How would the solution change if Hillary and Eugene ran in the same direction?
2. What is the relationship between the arc distance and linear distance in this context?
3. How can we generalize this problem for runners on any circular track?
4. What happens if one of the runners runs faster or slower than the given speeds?
5. How does the chord length formula apply to other geometric shapes?

Tip: When working with circles, remember that the chord length is always shorter than the arc length unless the runners are directly opposite each other.