Problem 314
The Mouse on the Moon
The moon has been opened up, and land can be obtained for free, but there is a catch. You have to build a wall around the land that you stake out, and building a wall on the moon is expensive. Every country has been allotted a 500 m by 500 m square area, but they will possess only that area which they wall in. 251001 posts have been placed in a rectangular grid with 1 meter spacing. The wall must be a closed series of straight lines, each line running from post to post.
The bigger countries of course have built a 2000 m wall enclosing the entire 250 000 m2 area. The Duchy of Grand Fenwick, has a tighter budget, and has asked you (their Royal Programmer) to compute what shape would get best maximum enclosed-area/wall-length ratio.
You have done some preliminary calculations on a sheet of paper. For a 2000 meter wall enclosing the 250 000 m2 area the enclosed-area/wall-length ratio is 125.
Although not allowed , but to get an idea if this is anything better: if you place a circle inside the square area touching the four sides the area will be equal to π*2502 m2 and the perimeter will be π*500 m, so the enclosed-area/wall-length ratio will also be 125.
However, if you cut off from the square four triangles with sides 75 m, 75 m and 75√2 m the total area becomes 238750 m2 and the perimeter becomes 1400+300√2 m. So this gives an enclosed-area/wall-length ratio of 130.87, which is significantly better.
Find the maximum enclosed-area/wall-length ratio.
Give your answer rounded to 8 places behind the decimal point in the form abc.defghijk.
月球上的老鼠
为了开发月球，现在可以免费地获得月球上的土地，但是且慢，你必须在你的土地周围建一堵墙才算数，而在月亮上建墙是非常昂贵的。现在每个国家都分配到一块500米乘500米的方形土地，但是最终只能持有用墙围起来的那部分。在每个国家的土地上，按照1米间隔打下了251001根桩，墙必须是由一系列直线构成的封闭曲线，每条直线都必须从一根桩开始到另一根桩结束。
当然，强大的国家会建一堵2000米长的墙，将全部250 000平方米都包围起来。但是大芬威克公国的预算有点紧张，因此要求你（皇家程序员）计算出什么形状可以使得包围面积和墙长的比值最大。
你在草稿纸上做了些初步的计算。用2000米的长围住全部的250 000平方米，包围面积和墙长的比值是125。
尽管并不允许这么做，但是考虑一下这种情况也无妨：如果你在方形土地内画一个圆和四条边相切，面积将会是π*2502平方米，周长则是π*500米，所以包围面积和墙长的比值仍然是125。
然而，如果你从方形土地的四个角各切掉一个边长为75米、75米和75√2米的三角形，总面积将会变成238750平方米，而周长变成1400+300√2米。因此包围面积和墙长的比值是130.87，比上述情况都要好得多。
求包围面积和墙长的最大比值。
将你的答案保留小数点后8位小数，即格式为abc.defghijk。