To Swim or to Run?

It is important to note that $\theta$ is measured in radians.

The first step is to draw a diagram!  You might also want to pick a variable to represent the speed at which Birkhoff Maclane can swim (such as $v$), and then you can write her running speed as $kv$.  You will also want to pick a variable to represent the diameter or radius of the pool (Radius might be easier). 

It might be easiest to start by considering just cases (i) and (iii). Can you find expressions for the length of time it will take her to swim across the pool, and the time it will take her to run around?  What is the optimum strategy if $k=1$?  What if $k=1000$?  Is there a value of $k$ where the times are the same?

When considering the case where she runs a bit and then swims, it is a good idea to check that when you substitute in $\theta = 0$ (when she swims straight across) or $\theta = {\pi} $ (when she runs around) you get the same answers as in cases (i) and (iii).

You might be able to simplify your expression for the time in case (ii) by considering double-angle trig formulae.

Can you use differentiation to find the minimum time it takes if she uses method (ii)? How can you check that this is a minimum?