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Somewhere Near the Center (Posted on 2008-09-09) Difficulty: 2 of 5
This problem was inspired by Charlie's "Continental Pole of Inaccessibility" problem.

An irregular two dimensional shape is depicted below. Using the unit markings find the following:

1. The "center of mass" for the shape.
2. The point on the shape that lies furthest from the edge.

8    +---+---+---+   +---+---+
     |           |   |       |
7+---+           +---+       +---+
 |                               |
6+                               +
 |                               |
5+                           +---+
 |                           | 
4+                           +---+
 |                               |
3+                           +---+
 |                           |
2+                           +---+
 |                               |
1+                       +---+---+
 |                       |
0+---+---+---+---+---+---+
 0   1   2   3   4   5   6   7   8

No Solution Yet Submitted by Leming    
Rating: 4.0000 (2 votes)

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Solution, part 2 Comment 3 of 3 |

To find the furthest point from the edge, one may inscribe the largest circle within the bounds of the shape. The center of that circle would then be the furthest point from the edge.

As it is, the largest circle that can be inscribed will include the point at cartesian coordinate (4,7). The distance from this point to the center of the circle, would then be equal to the radius of the circle, i.e., r = SQRT((4 - r)^2 + (7 - r)^2). Using the quadratic equation, the center is at
([11 - 2*SQRT(14)], [11 - 2*SQRT(14)]), or approximately
(3.51669, 3.51669).


  Posted by Dej Mar on 2008-09-09 19:11:07
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