Two cyclists start from the same point on a circular track of length 900 m at 7:00 a.m. and ride in opposite directions. Their speeds are 6 km/h and 9 km/h respectively. At what time will they meet for the fourth time?

Correct Answer: 7:14:24 a.m.

Step-by-Step Explanation:

Step 1: Understand the given parameters
- Total length of the circular track ($D$) = 900 m
- Speed of the first cyclist ($v_1$) = 6 km/h
- Speed of the second cyclist ($v_2$) = 9 km/h
- Start time = 7:00 a.m.
- Direction of movement = Opposite directions

Step 2: Convert speeds from km/h to m/s
To convert speed from km/h to m/s, multiply by $\frac{5}{18}$:

$v_1=6\times \frac{5}{18}=\frac{5}{3}\text{ m/s}$

$v_2=9\times \frac{5}{18}=\frac{5}{2}\text{ m/s}$

Step 3: Calculate the relative speed
Since the two cyclists move in opposite directions on the circular track, their relative speed is the sum of their individual speeds:

$\text{Relative Speed }\left(v_{\text{rel}}\right)=v_1+v_2=\frac{5}{3}+\frac{5}{2}=\frac{10+15}{6}=\frac{25}{6}\text{ m/s}$

Step 4: Find the time taken for the 1st meeting
The distance covered relative to each other for one meeting on a circular track is equal to the length of the track (900 m):

$\text{Time for 1st meeting }\left(t_1\right)=\frac{\text{Track Length}}{v_{\text{rel}}}=\frac{900}{25/6}=\frac{900\times 6}{25}=36\times 6=216\text{ seconds}$

Step 5: Find the total time taken for the 4th meeting
Since they meet every 216 seconds, the total time required to meet for the 4th time is:

$\text{Total Time }\left(T\right)=4\times 216=864\text{ seconds}$

Step 6: Convert total time from seconds into minutes and seconds
Dividing 864 seconds by 60 to get minutes and remaining seconds:

$864\text{ seconds}=14\text{ minutes and }24\text{ seconds}$

Step 7: Determine the exact meeting time
Adding 14 minutes and 24 seconds to the starting time of 7:00 a.m.:

$\text{Time of 4th meeting}=\text{7:00:00 a.m.}+\text{14 min 24 sec}=\text{7:14:24 a.m.}$

Thus, the two cyclists will meet for the fourth time at 7:14:24 a.m.