2.5 KiB
| id | title | challengeType | forumTopicId | dashedName |
|---|---|---|---|---|
| 5900f4a71000cf542c50ffb9 | Problem 314: The Mouse on the Moon | 5 | 301970 | problem-314-the-mouse-on-the-moon |
--description--
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 \text{m}^2 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 \frac{\text{enclosed-area}}{\text{wall-length}} ratio.
You have done some preliminary calculations on a sheet of paper. For a 2000 meter wall enclosing the 250 000 \text{m}^2 area the \frac{\text{enclosed-area}}{\text{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 \times {250}^2 \text{m}^2 and the perimeter will be π \times 500 \text{m}, so the \frac{\text{enclosed-area}}{\text{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\sqrt{2} m the total area becomes 238750 \text{m}^2 and the perimeter becomes 1400 + 300\sqrt{2} m. So this gives an \frac{\text{enclosed-area}}{\text{wall-length}} ratio of 130.87, which is significantly better.
Find the maximum \frac{\text{enclosed-area}}{\text{wall-length}} ratio. Give your answer rounded to 8 places behind the decimal point in the form abc.defghijk.
--hints--
theMouseOnTheMoon() should return 132.52756426.
assert.strictEqual(theMouseOnTheMoon(), 132.52756426);
--seed--
--seed-contents--
function theMouseOnTheMoon() {
return true;
}
theMouseOnTheMoon();
--solutions--
// solution required