Problem source

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: 
\begin{questionparts}
\item to swim directly to $B$. 
\item 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 $B$. 
\item to run round the pool from $A$ to $B$. 
\end{questionparts}
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 \textbf{(i)},
\textbf{(ii) }and \textbf{(iii) }she should choose for each value
of $k$. Is there one choice from \textbf{(i)}, \textbf{(ii) }and
\textbf{(iii) }she will never take whatever the value of $k$?