Problem:
Two distinct lines pass through the center of three concentric circles of radii , , and . The area of the shaded region in the diagram is of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: radians is degrees.)
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Answer Choices:
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Solution:
Let be the acute angle between the two lines. The area of shaded Region in the diagram is
The area of shaded Region is
The area of shaded Region is
Hence the total area of the shaded regions is . The area bounded by the largest circle is , so
Solving for gives .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions