To Swim or to Run?

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:

(i)  to swim directly to $B$.

(ii) 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 $B$.

(iii) 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 $k$?

There are some hints and suggestions for how to approach this problem in the Getting Started section.

Garret Birkhoff and Saunders Mac Lane were two American mathematicians, who together in 1941 wrote "A Survey of Modern Algebra" one of the first modern algebra textbooks for undergraduates.

STEP Mathematics II, 1995, Q5. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.