91 lines
6.3 KiB
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91 lines
6.3 KiB
HTML
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<h2>The Mouse on the Moon</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 314</h3><span style="width:300px;color:#666;">Published on Sunday, 12th December 2010, 07:00 am; Solved by 310</span></div>
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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.
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The bigger countries of course have built a 2000 m wall enclosing the entire 250 000 m<sup>2</sup> area. The <a href='http://en.wikipedia.org/wiki/Grand_Fenwick'>Duchy of Grand Fenwick</a>, has a tighter budget, and has asked you (their Royal Programmer) to compute what shape would get best maximum enclosed-area/wall-length ratio.
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You have done some preliminary calculations on a sheet of paper.
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For a 2000 meter wall enclosing the 250 000 m<sup>2</sup> area the
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enclosed-area/wall-length ratio is 125.<BR />
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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*250<sup>2</sup> m<sup>2</sup> and the perimeter will be π*500 m, so the enclosed-area/wall-length ratio will also be 125.
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However, if you cut off from the square four triangles with sides 75 m, 75 m and 75<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2 m the total area becomes 238750 m<sup>2</sup> and the perimeter becomes 1400+300<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2 m. So this gives an enclosed-area/wall-length ratio of 130.87, which is significantly better.
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Find the maximum enclosed-area/wall-length ratio.<BR />
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Give your answer rounded to 8 places behind the decimal point in the form abc.defghijk.
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