2005 AMC 10B Problem 8

Problem 8 of 25EasierGeometry

8

-foot by

10

-foot floor is tiled with square tiles of size

1

foot by

1

foot. Each tile has a pattern consisting of four white quarter circles of radius

\dfrac12

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?

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Solution

The four quarter circles on a tile together make one full circle of radius

\dfrac12,

with area

\pi\left(\dfrac12\right)^2 = \dfrac{\pi}{4}.

So each tile has shaded area

1 - \dfrac{\pi}{4}

square feet. There are

8 \cdot 10 = 80

tiles, so the total shaded area is

80\left(1 - \dfrac{\pi}{4}\right) = 80 - 20\pi.

Thus, A is the correct answer.

Tagged: circle area · area decomposition