Difference between revisions of "2001 AMC 12 Problems/Problem 10"
(New page: == Problem == The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to <math> \text{(A) }50 ...) |
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If the side of the small square is <math>a</math>, then the area of the tile is <math>9a^2</math>, with <math>4a^2</math> covered by squares and <math>5a^2</math> by pentagons. | If the side of the small square is <math>a</math>, then the area of the tile is <math>9a^2</math>, with <math>4a^2</math> covered by squares and <math>5a^2</math> by pentagons. | ||
| − | Hence exactly <math>5/9</math> of any tile are covered by pentagons, and therefore pentagons cover <math>5/9</math> of the plane. When expressed as a percentage, this is <math>55.\overline{5}\%</math>, and the closest integer to this value is <math>\boxed{ | + | Hence exactly <math>5/9</math> of any tile are covered by pentagons, and therefore pentagons cover <math>5/9</math> of the plane. When expressed as a percentage, this is <math>55.\overline{5}\%</math>, and the closest integer to this value is <math>56</math>. <math>\boxed{\mathrm{D}}</math> |
== See Also == | == See Also == | ||
{{AMC12 box|year=2001|num-b=9|num-a=11}} | {{AMC12 box|year=2001|num-b=9|num-a=11}} | ||
Revision as of 16:33, 23 August 2009
Problem
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
![[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } } [/asy]](http://latex.artofproblemsolving.com/2/c/8/2c8743395e4a086d2ae9a30a2d161157e7dc3077.png)
Solution
Consider any single tile:
![[asy] unitsize(1cm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; draw(p); [/asy]](http://latex.artofproblemsolving.com/3/6/f/36f7a20120732a8955c3f77ea0d1e81fc34cc6ea.png)
If the side of the small square is 
, then the area of the tile is 
, with 
covered by squares and 
by pentagons.
Hence exactly 
of any tile are covered by pentagons, and therefore pentagons cover of the plane. When expressed as a percentage, this is 
, and the closest integer to this value is 
. 
See Also
| 2001 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 9 |
Followed by Problem 11 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
