The famous film star Birkhoff Maclane is sunning herself by the side
of her enormous circular swimming pool (with centre \(O\)) at a point
\(A\) on its circumference. She wants a drink from a small jug of iced
tea placed at the diametrically opposite point \(B\). She has three
choices:
- to swim directly to \(B\).
- to choose \(\theta\) with \(0<\theta<\pi,\) to run round the pool to
a point \(X\) with \(\angle AOX=\theta\) and then to swim directly from
\(X\) to \(B\).
- to run round the pool from \(A\) to \(B\).
She can run \(k\) times as fast as she can swim and she wishes to reach
her tea as fast as possible. Explain, with reasons, which of
(i),
(ii) and
(iii) she should choose for each value
of \(k\). Is there one choice from
(i),
(ii) and
(iii) she will never take whatever the value of \(k\)?