Akshay and Bobby are running on a circular track of radius 147 metres. Bobby can complete a round in 84 seconds and the speed of Akshay is half the speed of Bobby. They started simultaneously towards each other from two points 294 metres diametrically opposite on the circular path. If they first meet at a point which is between the two points from where they started their race, after how much time from the start do they meet at that point for the fifth time?

To solve this problem, we need to calculate when Akshay and Bobby meet for the fifth time on the circular track.

### Step-by-Step Breakdown:

#### 1. Calculate Bobby's Speed:
The track's radius is \(147\) meters, so the circumference of the circular track is:
\[
\text{Circumference} = 2 \pi \times 147 = 294 \pi \text{ meters}
\]

Bobby completes one round in 84 seconds. Therefore, his speed is:
\[
\text{Speed of Bobby} = \frac{\text{Circumference}}{\text{Time}} = \frac{294 \pi}{84} = 3.5 \pi \text{ meters/second}
\]

#### 2. Calculate Akshay's Speed:
We are told that Akshay's speed is half of Bobby's speed. Thus, Akshay's speed is:
\[
\text{Speed of Akshay} = \frac{1}{2} \times 3.5 \pi = 1.75 \pi \text{ meters/second}
\]

#### 3. Combined Relative Speed:
Since they are running towards each other, their combined relative speed is the sum of their individual speeds:
\[
\text{Relative Speed} = 3.5 \pi + 1.75 \pi = 5.25 \pi \text{ meters/second}
\]

#### 4. Initial Distance Between Them:
They start from diametrically opposite points, so the initial distance between them is half the circumference of the track:
\[
\text{Initial Distance} = \frac{\text{Circumference}}{2} = \frac{294 \pi}{2} = 147 \pi \text{ meters}
\]

#### 5. Time to First Meeting:
The time for the first meeting is the time it takes for the combined relative speed to cover the initial distance:
\[
\text{Time for First Meeting} = \frac{\text{Initial Distance}}{\text{Relative Speed}} = \frac{147 \pi}{5.25 \pi} = 28 \text{ seconds}
\]

#### 6. Frequency of Meetings:
They will continue to meet after regular intervals. The time interval between consecutive meetings is when the relative distance they cover equals the circumference of the track. The time for each subsequent meeting is:
\[
\text{Time Between Consecutive Meetings} = \frac{\text{Circumference}}{\text{Relative Speed}} = \frac{294 \pi}{5.25 \pi} = 56 \text{ seconds}
\]

Thus, they will meet for the first time at \(28\) seconds, and then every \(56\) seconds thereafter.

#### 7. Time for Fifth Meeting:
The times for their meetings are as follows:
- 1st meeting: \(28\) seconds
- 2nd meeting: \(28 + 56 = 84\) seconds
- 3rd meeting: \(84 + 56 = 140\) seconds
- 4th meeting: \(140 + 56 = 196\) seconds
- 5th meeting: \(196 + 56 = 252\) seconds

Thus, Akshay and Bobby will meet for the fifth time after **252 seconds**.

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### Summary:
Akshay and Bobby meet for the fifth time after **252 seconds**.

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Do you want more details or have any questions?

Here are 5 related questions to explore:
1. What would happen if Akshay’s speed was faster than Bobby’s?
2. How many times would they meet in the first 10 minutes of running?
3. How would the result change if the track radius was doubled?
4. Can you generalize a formula to find when they meet for the \(n\)-th time?
5. How would their meeting pattern change if they ran in the same direction?

**Tip:** Always check the units of time and speed when calculating motion problems.

Solve the problem of Akshay and Bobby running on a circular track with a radius of 147 metres, where Bobby runs faster than Akshay. Using concepts of relative motion and circular motion, we calculate when they meet for the fifth time, including detailed step-by-step explanations of key formulas and calculations. Suitable for students in Grades 9-11.

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