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2005 AMC 12B Problems/Problem 5

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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?

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

Solution

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

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

See also

2005 AMC 12B Problems

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/2005_AMC_12B_Problems/Problem_5"

Want to learn how to tackle those tough AMC/AIME/Olympiad counting and probability problems? Check out Art of Problem Solving's Intermediate Counting & Probability by David Patrick.

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