The problem asks for the starting distance difference required for a runner in the outermost lane of a circular track to cover the same distance as a runner in the innermost lane.

Calculating Lane Radius Difference

There are 6 lanes, each 1 meter wide. The difference in the radius of the running path between the outermost lane and the innermost lane is determined by the number of lanes and their width.

• Number of lanes = 6
• Width of each lane = $1 \text{ m}$
• The difference in radius corresponds to the width of the lanes between the innermost and outermost lane. This is $(6 - 1)$ lanes.
• Radius difference $= (6 - 1) \times 1 \text{ m} = 5 \text{ m}$

Let $r_{in}$ be the radius of the path for the runner in the innermost lane. The radius for the runner in the outermost lane, $r_{out}$, is $r_{out} = r_{in} + 5 \text{ m}$.

Determining Circumference Difference

The distance covered in one lap is the circumference of the circular path. The formula for the circumference ($C$) of a circle is $C = 2 \pi r$.

• Circumference for the innermost lane: $C_{in} = 2 \pi r_{in}$
• Circumference for the outermost lane: $C_{out} = 2 \pi r_{out} = 2 \pi (r_{in} + 5 \text{ m})$
• The difference in distance covered per lap is $C_{out} - C_{in}$.
• Difference $= 2 \pi (r_{in} + 5) - 2 \pi r_{in}$
• Difference $= 2 \pi r_{in} + 10 \pi - 2 \pi r_{in}$
• Difference $= 10 \pi \text{ m}$

Finding the Starting Offset

The runner in the outermost lane covers $10 \pi$ meters more distance than the runner in the innermost lane for each lap completed around the same starting point. To cover the *same* distance, the runner in the outermost lane must start ahead. The distance they need to start ahead is exactly the difference in the distance they cover in a lap.

Therefore, the runner in the outermost lane should start $10 \pi$ meters ahead.