Anya and Disha are running on a jogging track, in the shape of a semi-circle, in the same direction. When Anya covers six rounds of the track, Disha only covers 3.5 rounds. If they start running in the opposite direction on the track simultaneously and from the same point, they will meet for the second time when Anya has covered 4km more than Disha. What is the length of the track?

Let the length of the jogging track be denoted as \( L \). We are given that:

- When Anya covers 6 rounds, Disha covers 3.5 rounds.
- When they start running in opposite directions, they will meet for the second time when Anya has covered 4 km more than Disha.

### Step 1: Determine the speed ratio of Anya and Disha
From the information that Anya completes 6 rounds while Disha completes 3.5 rounds in the same time, we can conclude that the ratio of their speeds is:

\[
\text{Speed of Anya} : \text{Speed of Disha} = 6 : 3.5 = \frac{6}{3.5} = \frac{12}{7}.
\]

Thus, the ratio of their speeds is \( 12 : 7 \).

### Step 2: Analyze their relative motion in opposite directions
When running in opposite directions, the relative speed between them is the sum of their individual speeds. Let the speed of Anya be \( 12x \) and the speed of Disha be \( 7x \), where \( x \) is some constant. Therefore, their relative speed is:

\[
\text{Relative speed} = 12x + 7x = 19x.
\]

### Step 3: Time to meet for the first and second time
They start running from the same point, so they will meet for the first time after they have covered a total distance equal to one complete round of the track. The time taken to meet for the first time is:

\[
t_1 = \frac{L}{19x}.
\]

For the second meeting, they will have covered a total distance of \( 2L \) (since they will meet again after covering two full rounds of the track). The time taken for this second meeting is:

\[
t_2 = \frac{2L}{19x}.
\]

### Step 4: Condition for the second meeting
We are told that by the time they meet for the second time, Anya will have covered 4 km more than Disha. In time \( t_2 \), the distances covered by Anya and Disha are:

- Distance covered by Anya: \( 12x \times t_2 = 12x \times \frac{2L}{19x} = \frac{24L}{19} \),
- Distance covered by Disha: \( 7x \times t_2 = 7x \times \frac{2L}{19x} = \frac{14L}{19} \).

The difference between their distances at the second meeting is:

\[
\frac{24L}{19} - \frac{14L}{19} = \frac{10L}{19}.
\]

This distance is given to be 4 km, so we have the equation:

\[
\frac{10L}{19} = 4.
\]

### Step 5: Solve for \( L \)
Solving for \( L \):

\[
L = \frac{4 \times 19}{10} = \frac{76}{10} = 7.6 \text{ km}.
\]

Thus, the length of the track is \( 7.6 \) km.

Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. If they run in the same direction, how long will it take for Anya to overtake Disha?
2. How many times would they meet if they ran for 30 minutes at the same speeds in opposite directions?
3. What if Disha increased her speed by 10%, how would that affect the time of their second meeting?
4. How much farther would Anya have run than Disha after 1 hour of running in opposite directions?
5. If Anya’s speed were reduced by 1 km/h, how would that change the time to their second meeting?

**Tip:** In relative motion problems, adding and subtracting speeds depending on the direction is key for finding when two objects meet or overtake.

Explore how to determine the length of a jogging track based on the running speeds of Anya and Disha. Using speed ratios and relative motion principles, learn how they meet when Anya covers 4 km more than Disha. Suitable for students in Grades 9-12.

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