Old Man River . . . In Figure 3.2.8(a) suppose that the y-axis and the dashed vertical l...

Old Man River . . . In Figure 3.2.8(a) suppose that the y-axis and the dashed vertical line x = 1 represent, respectively, the straight west and east beaches of a river that is 1 mile wide. The river flows northward with a velocity v_r, where mi/h is a constant. A man enters the current at the point (1, 0) on the east shore and swims in a direction and rate relative to the river given by the vector v_s, where the speed mi/h is a constant. The man wants to reach the west beach exactly at (0, 0) and so swims in such a manner that keeps his velocity vector v_s always directed toward the point (0, 0). Use Figure 3.2.8(b) as an aid in showing that a mathematical model for the path of the swimmer in the river is

[Hint: The velocity v of the swimmer along the path or curve shown in Figure 3.2.8 is the resultant v = v_s + v_r. Resolve v_s and v_r into components in the x- and y-directions. If are parametric equations of the swimmer’s path, then .]