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# Problem 262

Mountain Range

The following equation represents the continuous topography of a mountainous region, giving the elevation h at any point (x,y):

A mosquito intends to fly from A(200,200) to B(1400,1400), without leaving the area given by 0 ≤ x, y ≤ 1600.

Because of the intervening mountains, it first rises straight up to a point A’, having elevation f. Then, while remaining at the same elevation f, it flies around any obstacles until it arrives at a point B’ directly above B.

First, determine fmin which is the minimum constant elevation allowing such a trip from A to B, while remaining in the specified area.
Then, find the length of the shortest path between A’ and B’, while flying at that constant elevation fmin.

Note: For convenience, the elevation function shown above is repeated below, in a form suitable for most programming languages:
h=(5000-0.005*(x*x+y*y+x*y)+12.5*(x+y))*exp(-abs(0.000001*(x*x+y*y)-0.0015*(x+y)+0.7))

h=(5000-0.005*(x*x+y*y+x*y)+12.5*(x+y))*exp(-abs(0.000001*(x*x+y*y)-0.0015*(x+y)+0.7))