7 (Calculus of variations, Troutman 1.1.3) Assume a boat maintains a relative velocity V = (01, "2) with constant magnitude |v| = v while crossing a river of varying flow rate o c, = (0, o) from a point (z(0), y(0)) = (0,0) (2) to a point (x (T), y (T) ) = (L, Y). (3) This means the path of the boat is determined by the sum of these velocities according to the ODE dr dy de dt (4) (a) Show that the total time of transit for a path (x(t), y(t)) satisfying (2-4) is T = -dr. Hint(s): v = dx/dt. (Assume this quantity is non-vanishing so that the time of travel can be expressed as a function of r.) (b) If the path is expressed as the graph of a function y = u(x), write down an op- propriate admissible class and variational problem to determine the path giving the least time of travel across the river. Hint(s): Show that and solve this equation for 1/v. (c) What can you say about the sign of v- - 03p