Problem:
A circle of diameter is removed from a rectangle, as shown. Which whole number is closest to the area of the shaded region?
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Answer Choices:
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Solution:
The area of the circle is between and . To see this, draw squares around and inside the circle. The area of the large square is , the area of the small square is , and the circle fits between the two squares. The area of the rectangle with the circle removed is therefore between and , so the whole number closest to this area is .
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The area of the rectangle is , and the area of the circle with radius is , which is slightly larger than . Thus the area of the resulting figure is slightly smaller than , so it is closest to .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions