Narrow square paths ABCD and abcd have sides of $$200\text{ m}$$ and $$100\text{ m}$$. Two persons P and Q are initially located at a and A respectively. They both start walking in a clockwise sense with speeds of $$2\text{ m/s}$$ simultaneously.<br>When are they farthest from each other? (Give the first such time, in seconds after the start.)
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P goes round the inner square (perimeter 400 m) in 200 s, and Q goes round the outer square (perimeter 800 m) in 400 s. The greatest possible distance between points of the two paths is from an inner corner to the opposite outer corner, $$\sqrt{150^2 + 150^2} = 150\sqrt{2}\text{ m}$$. At $$t = 200\text{ s}$$, P is back at a and Q has gone half round to C, the opposite corner, so this first happens at 200 s (and then every 400 s, at 600 s, 1000 s and so on).