Two bike riders start together from the same point on a circular track and ride in opposite directions. Biker A rides at 16 km/h and biker B rides at 14 km/h. If the diameter of the track is 40 km, after how many hours will they first meet again?

Introduction / Context:
This problem involves relative speed on a circular track. Two bikers start from the same point at the same time but ride in opposite directions. We know their speeds and the diameter of the track. The goal is to find when they will first meet each other again after starting.

Given Data / Assumptions:

- Biker A speed = 16 km/h.
- Biker B speed = 14 km/h.
- The track is circular with diameter 40 km.
- Both riders start from the same point and move at constant speeds in opposite directions.

Concept / Approach:
On a circular track, the length of one full lap is the circumference. With diameter d, the circumference is pi * d. When the two riders move in opposite directions, their relative speed is the sum of their individual speeds. They will meet again when the total distance covered by both together equals exactly one full lap of the track. So we divide the circumference by the relative speed to get the time to first meeting.

Step-by-Step Solution:
Diameter of track = 40 km. Circumference C = pi * diameter = pi * 40 km. Combined (relative) speed when moving in opposite directions = 16 + 14 = 30 km/h. Time to meet again after the start = total distance for one lap / relative speed. So time t = C / 30 = (40 * pi) / 30 hours. Simplify: t = 4 * pi / 3 hours. Using pi approximately 3.14, t ≈ 4 * 3.14 / 3 ≈ 12.56 / 3 ≈ 4.19 hours. Rounded to two decimal places, t ≈ 4.18 hours (as in the options).

Verification / Alternative check:
A rough approximate check: The circumference is about 40 * 3.14 ≈ 125.6 km. At a combined speed of 30 km/h, time is 125.6 / 30 ≈ 4.19 hours. This matches the more precise calculation above and is very close to 4.18 hours, confirming the choice.

Why Other Options Are Wrong:
6.52, 8.14, and 5.02 hours: These times correspond to much larger or smaller distances when multiplied by the relative speed of 30 km/h and do not equal one full circumference of approximately 125.6 km. They represent incorrect interpretations of the relationship between distance, speed, and time on a circular track.

Common Pitfalls:
A common mistake is to forget that the two speeds must be added when riders move in opposite directions. Another error is to treat 40 km as the circumference rather than the diameter; that would underestimate the meeting time. Always compute the circumference as pi times diameter and use relative speed when two bodies move towards one another on a circular path.

Final Answer:
The two bikers will first meet again after approximately 4.18 hours.