Difference between revisions of "2005 AMC 12B Problems/Problem 5"
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== Solution == | == Solution == | ||
| + | There are 80 tiles. Each tile has <math>[\mbox{square} - 4 \cdot (\mbox{quarter circle})]</math> shaded. Thus: | ||
| + | |||
| + | <cmath> | ||
| + | \begin{align*} | ||
| + | \mbox{shaded area} &= 80 ( 1 - 4 \cdot (1/4) \cdot \pi \cdot (1/2)^2) \\ | ||
| + | &= 80(1-(1/4)\pi) \\ | ||
| + | &= \boxed{80-20\pi}. | ||
| + | \end{align*} | ||
| + | </cmath> | ||
== See also == | == See also == | ||
* [[2005 AMC 12B Problems]] | * [[2005 AMC 12B Problems]] | ||
Revision as of 22:34, 17 April 2009
Problem
An 
-foot by 
-foot floor is tiles with square tiles of size 
foot by foot. Each tile has a pattern consisting of four white quarter circles of radius 
foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
![[asy] unitsize(2cm); defaultpen(linewidth(.8pt)); fill(unitsquare,gray); filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black); filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black); filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black); filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black); [/asy]](http://latex.artofproblemsolving.com/c/b/e/cbe2bb4966beb1c1cbc889938c8e120b7d80639b.png)
Solution
There are 80 tiles. Each tile has ![$[\mbox{square} - 4 \cdot (\mbox{quarter circle})]$](http://latex.artofproblemsolving.com/6/1/7/617c9a81577193c333f0b80c64514805a1e55857.png)
shaded. Thus:
