The Lifeguard Problem

The following exercise appears in some version in almost every calculus textbook (e.g., [Fraleigh 1985, 1851; [Purcell and Varberg 1987, 174, 1771). A person standing on a straight lake shoreline wishes to reach a boat anchored in the lake at a point that is w meters out in the water (Figure 1). The w meters is measured from a point on the shore that is d meters from the person. The person can run r meters per second and can swim s meters per second.