Be the one that minimizes the time of flight, t, which was responsible for snell's law in elementary optics, for a medium with a constant index of refraction, such that the angle between the two straight paths was discovered. to extend this concept dramatically, consider the situation that the light travels in a general medium where the index of refraction is n, which depends on the specific path, where the speed of light in such a region of space is v=c/n, and c is the speed of light. for the differential path length as , where v = d/dt , the differential time of flight is dnds. for a general path, y(x). in 2d space, where the light travels along the axis, as it increases along the y axis, the time differential can be expressed. as usual, in terms of a spatial differential, d =f(x, y,a)dr, so that the time of flight is given as the functional t-f(v. y,x)dr. the objective is to solve the variational calculation to determine the specific path the light takes, y(x), which minimizes the time of flight functional, for the situation that the index of refraction in the medium is n=vyz +1 , for the case where the path goes from the (x, y) coordinates of (0,1) to (1,e). recall that the minimum of a functional, t. can be determined by the stationary solution of the variation of the functional, δτ-0, which results in the euler equation, 0, and if the integrand, f(. of the f depend on r, the euler equation solution to the variational problem is most easily achieved with the alternate euler equation, ,f-y, k, where the constant k can be chosen to satisfy the boundary conditions. a) determine the explicit time functional, t, for the light travel time in this medium, as a definite integral over the r coordinate, using an unknown path function, y(x),as well as to give the precise integrand, f(v. y,.x). b) using the appropriate euler equation, determine the path, y(x), which minimizes the time functional, t, and thus satisfies fermat's principle. note that, after you reduce the euler equation to an integral over x and y, it will be best to chose the constant k value which will allow you to easily perform the integration, which will also allow you to satisfy the boundary conditions that are given: (0,1). (l. e). e) given your path result, sketch the light path which satisfies fermat's principle, for the index of refraction that was utilied above, n-v+. consequently, you should be able to describe why your solution path, y(x), might provide a shorter time of light travel than a simple straight line path (which is, of course, the shortest distance path). by,