Final answer:
The value of x which indicates the frequency of trains leaving the endpoint is every 6 minutes, because the dog meets a train every 4 minutes and gets caught by a train every 12 minutes.
Step-by-step explanation:
We need to calculate the value of x which represents the frequency in minutes with which trains leave the endpoint of a route given that a dog on the tracks encounters a train every 4 minutes and a train catches up to him every 12 minutes.
The fact that a train meets the dog every 4 minutes implies that this is the time it takes for two consecutive trains to pass the same point, which means it is the minimum possible value for x. That is because the trains cannot be departing more frequently than the time it takes for them to pass a point.
The fact that a train catches up to the dog every 12 minutes implies that there are 3 intervals of 4 minutes within the 12-minute period. The situation described is only possible when the trains are dispatched at intervals that are a multiple of the 4-minute cycle. Since the 12-minute catch-up period is equal to three times the 4-minute meeting interval, the possible dispatch interval, x, must be a factor of 12.
Out of the available options, the correct dispatch interval that satisfies both conditions and is a factor of 12 is every 6 minutes, which designates option c) as the correct answer.