Difference between revisions of "2011 AMC 8 Problems/Problem 25"
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==Solution== | ==Solution== | ||
+ | ===Solution 1=== | ||
+ | The area of the smaller square is the one half of the product of its diagonals. Note that the distance from a corner of the smaller square to the center is equivalent to the circle's radius so the diagonal is equal to the diameter: <math>2*2*1/2=2.</math> | ||
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+ | The circle's shaded area is the area of the smaller square subtracted from the area of the circle: <math>\pi - 2.</math> | ||
+ | |||
+ | If you draw the diagonals of the smaller square, you will see that the larger square is split <math>4</math> congruent half-shaded squares. The area between the squares is equal to the area of the smaller square: <math>2.</math> | ||
+ | |||
+ | Approximating <math>\pi</math> to <math>3.14,</math> the ratio of the circle's shaded area to the area between the two squares is about | ||
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+ | <cmath>\frac{\pi-2}{2} \approx \frac{3.14-2}{2} = \frac{1.14}{2} \approx \boxed{\textbf{(A)}\ \frac12}</cmath> | ||
+ | |||
+ | ===Solution 2=== | ||
+ | For the ratio of the circle's shaded area to the area between the squares to be <math>1,</math> they would have to be approximately the same size. For any ratio larger than that, the circle's shaded area must be greater. However, we can clearly see that the circle's shaded area is part of the area between the squares, and is approximately <math>\boxed{\textbf{(A)}\ \frac12}</math> | ||
==See Also== | ==See Also== | ||
− | {{AMC8 box|year=2011|num-b=24| | + | {{AMC8 box|year=2011|num-b=24|after=Last Problem}} |
Revision as of 19:00, 25 November 2011
Problem
A circle with radius is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?
Solution
Solution 1
The area of the smaller square is the one half of the product of its diagonals. Note that the distance from a corner of the smaller square to the center is equivalent to the circle's radius so the diagonal is equal to the diameter:
The circle's shaded area is the area of the smaller square subtracted from the area of the circle:
If you draw the diagonals of the smaller square, you will see that the larger square is split congruent half-shaded squares. The area between the squares is equal to the area of the smaller square:
Approximating to the ratio of the circle's shaded area to the area between the two squares is about
Solution 2
For the ratio of the circle's shaded area to the area between the squares to be they would have to be approximately the same size. For any ratio larger than that, the circle's shaded area must be greater. However, we can clearly see that the circle's shaded area is part of the area between the squares, and is approximately
See Also
2011 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |