This is the first of two articles devoted to the International Year of Light.
Let us suppose that in a piece of coast line, straight and without currents, a swimmer asks for help and that the lifeguard, on the beach, goes towards him to help. What path should the lifeguard choose so that he would spend the least possible time to get to the swimmer? Naturally, the answer depends on the running velocity, on the sand, and the swimming velocity, in the sea. If they are equal, an improbable scenario, the fastest path coincides with the shortest path and, it follows, is the segment of line that connects the initial position of the lifeguard to the swimmer. But, in general, one can run much faster than one can swim. Even considering this observation, there is a case where the answer is the same as before: when the segment of line that connects the lifeguard and the swimmer is perpendicular to the coast line. What about the other cases? A little bit of common sense tells us that one should increase the path length in the sand, where one can achieve high velocities, and reduce the path length in the water, since people are usually slower at swimming.
But how can we determine, with precision, the best point \(C\) in the coast line, where we can make the transition from running to swimming? Another question: we have admitted that the lifeguard was on the beach when the swimmer asked for help. At first, one might think that, if he is already in the water, the best option will be swimming in a straight line. But is this necessarily the best option?
This text is a slightly modified version of the following article (in Portuguese) published by Atractor in Gazeta de Matemática
(*) The interactive aplications included throughout this text were developed under a grant by FCT - Fundação para a Ciência e a Tecnologia.
Difficulty level: Upper Secondary, University