Optimization using Fermat's principle. Help!

Question

anonymous · Answer

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anonymous · Answer

How do i start with this type of question?

xishem · Answer

Whenever you have an optimisation problem you want to find some function for the value you want to minimise and then go from there. So can you find a function for d?

kainui · Answer

A good way to start these kinds of problems is to just sort of draw out the figure on your own paper and label the sides of the triangles and the two angles that are important here. Now make all the formulas you possibly can from that information using sine, cosine, and pythagorean theorem. Then see if you can piece the equations together by plugging some into the others to get something useful. The point is, once you start playing around with it you'll start to see, but just sitting looking at it will lead nowhere.

If that doesn't work, I'll check up on you in about 30 minutes, but I won't help unless I've seen a good faith effort. Good luck!

anonymous · Answer

|dw:1385723355651:dw|
$$L=\sqrt{m^2+n^2}+\sqrt{n^2+(d-x)^2}$$

anonymous · Answer

i differentiate it and got a weird expression which is dont know how to relate with the angles. $$L'(x)=\frac{ x }{ \sqrt{m^2+x^2} }=\frac{ d-x }{ \sqrt{n^2+(d-x)^2} }$$

anonymous · Answer

did you use an i phone app to make that picture

anonymous · Answer

huh? no i used the draw button, there is a straight line tool that stretches. i used a regular mouse making that btw.