A lifeguard stands at point L on a straight beach. A swimmer in distress is located at point S, which is 80m along the shoreline from L and 50m out to sea (perpendicular to the shore). The lifeguard can run at 6m/s on the sand and swim at 1.5m/s in the water. She may choose any point P on the shoreline between L and the point directly opposite S to enter the water, as shown in the diagram below (no diagram is provided; assume a right-angle layout).
a. Part 1: (a) Let x be the distance in meters from L to the point P where the lifeguard enters the water (0≤x≤80). Write an expression for the total rescue time T(x) in terms of x. Specify the domain for T.
b. Part 2: (b) Determine all critical points of T on the domain from part (a). Show explicit computation of the first derivative and algebraic simplification leading to the critical point(s).
c. Part 3: (c) Establish which critical point minimizes the rescue time by examining the sign of T′(x) on each relevant interval or by another valid justification. State the optimal value of x and the corresponding minimum rescue time.
d. Part 4: (d) Comment on how the optimal x would change if the lifeguard’s swimming speed increased while the running speed remained the same. Provide a qualitative explanation without additional calculus.