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Problem 262


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.

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))


山的范围

如下的等式表示了一座山连续的地貌,对于任意一点(x,y),其海拔h为:

一只蚊子试图从点A(200,200)飞往点B(1400,1400),但不能离开0 ≤ x, y ≤ 1600的这片区域。

为了避免地貌的影响,它先竖直向上飞到一点A’,其还把为f。然后,它保持海拔高度f不变,绕过所有障碍,直到它到达B点正上方的一点B’。

首先,找出在给定区域内存在由A至B路径的最小海拔高度fmin
其次,找出A’到B’的最短路径长度,路径上的海拔高度始终为常数fmin

给出最短路径长度并四舍五入至三位小数作为你的答案。

注意:方便起见,上述海拔函数在下面按照方便大多数编程语言习惯的方式重写为:
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))