1. The problem statement, all variables and given/known data A rectangular solid of height h increases in density as its height increases, so the index of refraction of the solid increases with height according to: n(y) = 1.20(2y + 1) where y is the distance, in meters, from the origin (see diagram). A beam of light traveling in air (n = 1.00) in the x-y plane strikes the bottom of the tank at the origin, making an angle of incidence with the normal of θ1. Assume: • n varies only with y, not with x or z. • The light travels in the x-y plane. • The block is wide, so the light leaves through the top and not through a side. a) Use Fermat’s principle and Euler’s equation to find the equation for the path of the light y(x) in the tank. There will be two undetermined constants. HINT: If the equations become too complicated (and they will), remember we learned some alternative ways to solve calculus of variations problems. HINT: The answer should come out as a hyperbolic function. 2. Relevant equations Euler's equations for extremum: f - y'(∂f/∂y') = constant n=c/v 3. The attempt at a solution I'm looking to use Fermat's least time principle, I've tried to find the minimum of an integrated functional equal to time t. Currently I have it as: t = ∫ ds/v = ∫(1.20/c)(2y+1)√(1+y'^2)dx which would make my functional: f=(2y+1)√(1+y'^2) and using the Euler equations above I have from that: (2y+1)(1+y'^2)^.5 - y'^2(2y+1)(1+y'^2)^-.5 = (2y+1)(1+y'^2)^-.5 = constant and that's where I'm stuck... (2y+1)(1+y'^2)^-.5 = constant seems like it should be easy enough to integrate with dy and dx somehow, but try as I might I'm totally stumped. Somehow I'm supposed to get a hyperbolic function for y(x), which leads me to believe I'm approaching this whole problem the wrong way. Please help?