The Lifeguard Problem
Calculus
A classic problem with many personal takes - Explore the mathematical principles and create your own unique investigation.
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
How would you model the crossing of a body of water like Lake Geneva (Lac Léman) or the Mediterranean Sea? They're called optimal path problems, do your research too.
Guiding Questions
- A lifeguard runs faster on sand than they swim. Where should they enter the water to reach a swimmer fastest?
- Set up total time as a function of the entry point and minimise it. If you're doing AI SL, do this graphically or with technology; if you have the chain rule (AI HL), differentiate directly and compare.
- How does the best path change with the speed ratio? Connect it to how light bends between air and water.
- Make it personal: measure your own run and swim speeds and compute your optimal path for a real beach geometry.