Q1.
What is the connection between this question and the one about the 400 m athletics track?
Answer
They are the same question. The bends of the track are semicircles, so the length of a lap is two straights plus one full circle — and you cannot compute that until you know how to find the length round a circle.
Lap length = 2 × 84.39 + 2πr
Stagger between lanes = 2π(r + 1.22) − 2πr = 2π × 1.22
Stagger between lanes = 2π(r + 1.22) − 2πr = 2π × 1.22
Both expressions contain π. Without a value for the C/D ratio the track designer cannot mark a single starting line.
Why it happens: the track problem is a real-world problem and the circle problem is a mathematical one, but the real-world problem was solved only after the mathematical one was. This is a pattern you will see again and again: a practical need (laying out fields, storing grain in round towers, taxing circular plots) pushes people to answer a purely geometric question.