2006 AMC 10B Problem 4

Problem 4 of 25EasierGeometry

Circles of diameter

1

inch and

3

inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?

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Solution

The red circle has area

\pi(\tfrac12)^2 = \tfrac{\pi}{4},

and the large circle has area

\pi(\tfrac32)^2 = \tfrac{9\pi}{4}.

The blue ring is

\tfrac{9\pi}{4}-\tfrac{\pi}{4}=2\pi.

The ratio is

2\pi \div \tfrac{\pi}{4} = 8.

Thus, the correct answer is D.

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