A lifeguard stands 60 m from the shoreline of a straight beach at point L. A swimmer in distress is located at point S, which is 30 m offshore and 80 m farther along the shoreline from the point on the shore closest to L. The lifeguard can run at 8 m/s on sand and swim at 2 m/s in water. Assume the lifeguard travels in a straight line while running, enters the water at point P on the shoreline, and then swims in a straight line to S.
Let x denote the distance, measured along the shoreline from the point closest to L, to the chosen entry point P (0 ≤ x ≤ 80).
(a) Express the total rescue time T(x) as a function of x.
(b) Determine dxdT and identify all critical points of T on the interval [0, 80].
(c) Find the value of x that minimizes the total rescue time and compute this minimal time to the nearest tenth of a second.
(d) Justify that the critical point found in part (b) corresponds to a minimum on the closed interval.
(e) Interpret your result by stating how far the lifeguard should run along the beach before entering the water.