I'm not sure of where to begin with this question? completely lost

Two cyclists start out on the same trail together that is overall 20km long. The first cyclist completes the entire trail and is traveling at a constant 20 km/hr for the away part of the trip and then only 15 km/hr for the return trip. the Second cyclist, who is slower, is moving along at a Constant 10 km/hr for the entire trip. The second cyclist wants to turn around at some point so that he will arrive back at the starting position at the same time that the first cyclist will arive. At what point along the path should he do that?

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To solve this problem, we need to analyze the time it takes for both cyclists to complete the trail.

Let's start with the first cyclist. The total distance is 20 km, and the speed for the away part of the trip is 20 km/hr. Therefore, the time taken for the away part of the trip is:

time_away = distance / speed_away = 20 km / 20 km/hr = 1 hour

For the return trip, the speed is 15 km/hr. So, the time taken for the return trip is:

time_return = distance / speed_return = 20 km / 15 km/hr = 4/3 hours = 1.33 hours

Hence, the total time taken by the first cyclist is:

total_time_first = time_away + time_return = 1 hour + 1.33 hours = 2.33 hours

Now let's look at the second cyclist, who is traveling at a constant speed of 10 km/hr. The total time taken by the second cyclist to complete the entire trail is:

total_time_second = distance / speed_second = 20 km / 10 km/hr = 2 hours

To find the point along the trail where the second cyclist should turn around, we need to find the distance the second cyclist should travel before turning around. Let's assume this distance is x km.

The time taken by the second cyclist to cover the distance x km is:

time_second = x km / 10 km/hr = x/10 hours

Since we want both cyclists to arrive back at the starting position at the same time, we need to determine when the total time taken by the first cyclist is equal to the total time taken by the second cyclist.

total_time_first = total_time_second

2.33 hours = 2 hours + (x/10) hours

0.33 hours = x/10 hours

0.33 * 10 = x

3.3 km = x

Therefore, the second cyclist should turn around at a distance of 3.3 km along the trail in order to arrive back at the starting position at the same time as the first cyclist.