Question 13

Explanation

Analyzing the Track

The track is circular with the checkpoint sequence $A \to B \to C \to D \to A$. Since the distance between checkpoints is equal, we can count the number of segments the runner has traversed. Starting from $A$, the runner has passed checkpoint $C$ $3$ times.

1. Reaching $C$ for the 1st time: $A \to B \to C$ ($2$ segments).
2. Reaching $C$ for the 2nd time: $C \to D \to A \to B \to C$ ($4$ segments).
3. Reaching $C$ for the 3rd time: $C \to D \to A \to B \to C$ ($4$ segments).

The total segments traversed is the sum of these stages.

Calculating Time per Segment

It is known that the time required to cover $10$ segments is $12$ minutes. We can determine the time taken for one segment.

Calculating Total Time to Reach C for the 4th Time

To reach checkpoint $C$ for the fourth time, the runner must complete another full lap from their last position (at the 3rd $C$). The additional path is $C \to D \to A \to B \to C$, which consists of $4$ segments.

The additional time required is:

Thus, the total time from the start to reaching checkpoint $C$ for the fourth time is the sum of the initial time and the additional time.

Therefore, the time required is $16.8$ minutes.