Problem 7 (Calculus of variations, Troutman 1.1.3) Assume a boat maintains a relative velocity V = (V1, V2) with constant magnitude |v| = v while crossing a river of varying flow rate o ez = (0, 6) 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 dy It =v+pez. (4) (a) Show that the total time of transit for a path (x(t), y(t)) satisfying (2-4) is T = L d.r . V1 Hint(s): 1 = dx/dt. (Assume this quantity is non-vanishing so that the time of travel can be expressed as a function of x.) (b) If the path is expressed as the graph of a function y = u(x), write down an ap- propriate admissible class and variational problem to determine the path giving the least time of travel across the river. Hint(s): Show that V1 and solve this equation for 1/v. (c) What can you say about the sign of v2 - 62