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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 by parts (spoiler) | Comment 1 of 3

well I started by braking it up into 8 rectangular areas.  for each of these I computed A (area of rectangle), x (x-coord of centroid of rectangle), and y (y-coord of centroid of rectangle).   To get the rectangles I simply started at the top left worked my way right and down.  For ease of notation when I reference a rectangle I will give its dimensions axb and the coordinate of its top left vertice.  So the first rectangle is the 3x1 starting at (1,8).  next is a 2x1 at (5,8) then a 8x2 at (0,7).  I think you get the idea of my system.  I get the following table of value

<TABLE style="WIDTH: 288pt; BORDER-COLLAPSE: collapse" cellSpacing=0 cellPadding=0 width=384 border=0 x:str> <COLGROUP> <COL style="WIDTH: 48pt" span=6 width=64> <TBODY> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; WIDTH: 48pt; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" width=64 height=17>n</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; WIDTH: 48pt; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" width=64>An</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; WIDTH: 48pt; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" width=64>xn</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; WIDTH: 48pt; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" width=64>yn</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; WIDTH: 48pt; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" width=64>xn*An</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; WIDTH: 48pt; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" width=64>yn*An</TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17 x:num>0</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>3</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>2.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>7.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=C2*B2">7.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=D2*B2">22.5</TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17 x:num>1</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>2</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>6</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>7.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=C3*B3">12</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=D3*B3">15</TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17 x:num>2</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>16</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>4</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>6</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=C4*B4">64</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=D4*B4">96</TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17 x:num>3</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>7</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>3.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>4.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=C5*B5">24.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=D5*B5">31.5</TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17 x:num>4</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>8</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>4</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>3.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=C6*B6">32</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=D6*B6">28</TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17 x:num>5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>7</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>3.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>2.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=C7*B7">24.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=D7*B7">17.5</TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17 x:num>6</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>8</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>4</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>1.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=C8*B8">32</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=D8*B8">12</TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17 x:num>7</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>6</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>3</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num>0.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=C9*B9">18</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=D9*B9">3</TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17>total</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=SUM(B2:B9)">57</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent"></TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent"></TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=SUM(E2:E9)">214.5</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num x:fmla="=SUM(F2:F9)">225.5</TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17>cx</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num="3.763157894736842" x:fmla="=E10/B10">3.763158</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent"></TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent"></TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent"></TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent"></TD></TR> <TR style="HEIGHT: 12.75pt" height=17> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; HEIGHT: 12.75pt; BACKGROUND-COLOR: transparent" height=17>cy</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent" x:num="3.9561403508771931" x:fmla="=F10/B10">3.95614</TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent"></TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent"></TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent"></TD> <TD class=xl22 style="BORDER-RIGHT: #f0f0f0; BORDER-TOP: #f0f0f0; BORDER-LEFT: #f0f0f0; BORDER-BOTTOM: #f0f0f0; BACKGROUND-COLOR: transparent"></TD></TR></TBODY></TABLE>

where An is the area, xn is the x-coord of centroid, yn is y-coord of centroid.  Now its quite simple to use excel to compute overall cetroid.  fo x-coord simply take the sum An*xn for each n then divide by total area.  do same for y-coord except with An*yn.  this gives centroid of (3.763158,3.95614)

 

As for the second question, it is not so simple to solve I am seeing so I will continue to work on it


  Posted by Daniel on 2008-09-09 13:48:21
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