MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A lifeguard stands at point L on a straight beach. A swimmer in distress is at point S in the water, 90 m directly offshore and 200 m east of L. The lifeguard can run at 7.5 m/s on sand and swim at 1.6 m/s in the water.
She will run from L to some point R on the water’s edge, enter the water at R, and then swim in a straight line to S. Assume the shoreline is a straight east–west line and that the water depth changes abruptly at this line so both speeds remain constant.
Your task is to determine the optimal entry point R that minimizes the total rescue time.
(a) Let x represent the eastward distance (in meters) from L to R. Derive an expression for the total rescue time T(x) in terms of x and specify the domain of definition.
(b) Find T′(x) and determine all critical numbers of T within the domain identified in part (a).
(c) Prove that the critical number you have found produces the absolute minimum rescue time on the domain. Provide a calculus-based justification.
(d) Compute the minimizing value of x (to the nearest tenth of a meter) and the corresponding minimum rescue time (to the nearest tenth of a second). State both results clearly, including appropriate units.
(e) Explain, in the context of the problem, how the lifeguard’s differing speeds on sand and in water influence the location of the optimal entry point.