Chapter 2, Problem 97GP

(III) A lifeguard standing at the side of a swimming pool spots a child in distress, Fig. 2–53. The lifeguard runs with average speed ${\upsilon }_{\text{R}}$ along the pool’s edge for a distance x, then jumps into the pool and swims with average speed ${\upsilon }_{\text{S}}$ on a straight path to the child, (a) Show that the total time t it takes the lifeguard to get to the child is given by

   $t=\frac{x}{{\upsilon }_{\text{R}}}+\frac{\sqrt[]{D^2+{(d-x)}^2}}{{\upsilon }_{\text{S}}}$.

(b) Assume ${\upsilon }_{\text{R}}$ = 4.0 m/s and ${\upsilon }_{\text{S}}$ = 1.5m/s. Use a graphing calculator or computer to plot t vs. x in part (a), and from this plot determine the optimal distance x the life-guard should run before jumping into the pool (that is, find the value of x that minimizes the time t to get to the child).