A jogger starts from one end of a 13-mile nature trail at 8:00 A.M. One hour later, a cyclist starts from the other end of the trail and rides toward the jogger. If the rate of the jogger is 6 mph and the rate of the cyclist is 9 mph, at what time will the two meet?
Let the time until they meet be t hours
6t + 9t = 13
15t = 13
t = 13/15 hours = 52 minutes
they will meet at 8:52 am
To find the time when the jogger and the cyclist will meet, we need to determine how long it takes for them to cover the entire trail.
Let's first calculate the time it takes for the jogger to reach the meeting point. The jogger's speed is 6 mph, and the distance from the jogger to the meeting point is half of the trail's length, which is 13 miles / 2 = 6.5 miles.
Using the formula time = distance / speed, we can find the time taken by the jogger:
time = 6.5 miles / 6 mph
time = 1.08 hours
Next, we need to account for the fact that the jogger starts one hour earlier than the cyclist. Therefore, the total time taken for the cyclist to reach the meeting point is 1.08 hours - 1 hour = 0.08 hours.
To convert this time to minutes, we multiply by 60:
0.08 hours * 60 minutes/hour = 4.8 minutes
Therefore, it will take the cyclist 4.8 minutes to reach the meeting point.
Now, we can add the cyclist's time to the starting time of the jogger to find the meeting time:
8:00 A.M. + 4.8 minutes = 8:04.8 A.M.
Hence, the jogger and the cyclist will meet at approximately 8:04.8 A.M.
To find the time at which the jogger and cyclist meet, we need to determine the time it takes for them to cover the distance between them.
Let's start by finding how far the jogger travels in one hour. The rate of the jogger is given as 6 mph. Therefore, the distance the jogger covers in one hour is:
Distance = Rate × Time
Distance = 6 mph × 1 hour = 6 miles
Since the jogger has covered 6 miles in the first hour and the total distance of the trail is 13 miles, the remaining distance between them is:
13 miles - 6 miles = 7 miles
Now, let's calculate how long it takes for the cyclist to travel this 7-mile distance. The rate of the cyclist is given as 9 mph. Therefore, the time taken by the cyclist to cover 7 miles is:
Time = Distance / Rate
Time = 7 miles / 9 mph = 0.78 hours (rounded to two decimal places)
We know that the cyclist started one hour later than the jogger. So, the total time passed is the time taken by the jogger in the first hour plus the time taken by the cyclist to cover the remaining distance:
Total Time = 1 hour + 0.78 hours = 1.78 hours
Now, we need to determine the time at which the two will meet, which is 8:00 A.M. (the jogger's starting time) plus the total time it took for them to meet:
Meeting Time = Starting Time + Total Time
Meeting Time = 8:00 A.M. + 1.78 hours
To express the time in hours and minutes, we can convert 0.78 hours to minutes by multiplying it by 60:
0.78 hours × 60 minutes/hour = 46.8 minutes
Therefore:
Meeting Time = 8:00 A.M. + 1 hour and 46.8 minutes
Meeting Time = 9:46.8 A.M.
So, the jogger and cyclist will meet at approximately 9:46 A.M.