To Swim or to Run?

Why do this problem?

This problem uses ideas of arc length and chord lengths, as well as possibly differentiation or graph sketching to justify the final results (which might be surprising!).

Possible Approach

This problem featured in an NRICH Secondary webinar in March 2022.

You may want to ask students to consider Where to Land before introducing them to this problem.

Start by just considering methods (i) and (iii).

Ask students for expressions for the time it takes Birkhoff to swim directly across and to run all the way around.  They will need to choose some variables for speed and either radius or diameter.  Can they work out the values of $k$ for which running is the better option?

Method (ii) - if Birkhoff decides to run part of the way, and then swim, can they find expressions for the length that she runs and swims?  What is the total time taken for this method?  

How could we check our expression for the time taken in method (ii)?
Substituting $\theta = 0$ and $\theta = \pi$ should give the same answers as for methods (i) and (iii).

Can we find the minimum time taken with method (ii)?  How can we be sure it is a minimum?

Possible Extension

What happens if the drink is not diametrically opposite?  What if it is $\frac 1 3$ or $\frac 1 4$ of the way around the circle?

Note

Garrett Birkhoff and Saunders Mac Lane published their Survey of Modern Algebra in 1941. The book was written because the authors could find no adequate text to use with their students at Harvard.