2005 AMC 12B Problems/Problem 5

Problem

An $8$ -foot by $10$ -foot floor is tiles with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $1/2$ 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?

$[asy] unitsize(2cm); defaultpen(linewidth(.8pt)); fill(unitsquare,gray); filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black); filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black); filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black); filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black); [/asy]$

$\mathrm{(A)}\ 80-20\pi \qquad \mathrm{(B)}\ 60-10\pi \qquad \mathrm{(C)}\ 80-10\pi \qquad \mathrm{(D)}\ 60+10\pi \qquad \mathrm{(E)}\ 80+10\pi$

Solution

There are 80 tiles. Each tile has $[\mbox{square} - 4 \cdot (\mbox{quarter circle})]$ shaded. Thus:

$\begin{align*} \mbox{shaded area} &= 80 ( 1 - 4 \cdot (1/4) \cdot \pi \cdot (1/2)^2) \\ &= 80(1-(1/4)\pi) \\ &= \boxed{80-20\pi}. \end{align*}$

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2005 AMC 12B Problems/Problem 5

Problem

Solution

See also