A long distance runner starts at the beginning of a trail and runs at a rate of 5 miles per hour. One hour later, a cyclist starts at the beginning of the trail and travels at a rate of 13 miles per hour. What is the amount of time that the cyclist travels before overtaking the runner?
Do not do any rounding.
when the cyclist starts, the runner is 5 miles ahead.
So, he needs to make up 5 miles at 13 mi/hr, which takes 5/13 hours.
5/8
To determine the amount of time the cyclist travels before overtaking the runner, we can use the concept of relative speed.
First, let's assume that the time it takes for the cyclist to overtake the runner is "t" hours.
Since the runner has a one-hour head start, the runner has already traveled a distance of 5 miles (as the runner is running at a rate of 5 miles per hour).
To find out the distance between the cyclist and the runner when the cyclist overtakes the runner, we can multiply the cyclist's speed by the time it takes for the cyclist to overtake the runner (i.e., 13t).
We know that when the cyclist overtakes the runner, the distance traveled by the cyclist is equal to the distance traveled by the runner plus the initial distance the runner covered before the cyclist started.
Thus, we can write the equation:
13t = 5 + 5t
Simplifying the equation, we get:
13t = 5 + 5t
13t - 5t = 5
8t = 5
t = 5/8
Therefore, the cyclist will travel for 5/8 hours before overtaking the runner.