## Mountain Range

### Problem 262

Published on Friday, 30th October 2009, 09:00 pm; Solved by 475; Difficulty rating: 80%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 `f _{min}` 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

`f`.

_{min}Give that length as your answer, rounded to three decimal places.

__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) )