MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A lifeguard is standing on a straight, flat beach at point L, which lies 90 m east of the point S on the shore that is closest to a swimmer in distress. The swimmer is located 60 m due south of point S in the water. The lifeguard can run at a constant speed of 8 m/s on sand and swim at a constant speed of 2 m/s in the water. The lifeguard chooses a point P on the shoreline between L and S, located x meters east of S (so that x = 0 at S and x = 90 at L), at which to enter the water and then swim directly to the swimmer.
Assume the lifeguard immediately changes medium at P (no delay) and that the running and swimming speeds remain constant throughout the rescue.
(a) Express the total rescue time T(x) as a function of the chosen shore-entry distance x.
(b) Determine the value of x that minimizes T(x). Show all calculus work, including verification that your critical point corresponds to a minimum.
(c) Interpret the minimizing value of x in the physical context of the problem. Explain why the optimal entry point is not necessarily directly opposite the swimmer.
(d) Compute the minimum time (to the nearest 0.01 s) required for the lifeguard to reach the swimmer using your optimal x. Include units in your answer.