1987 AJHSME Problems/Problem 22

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Problem

$\text{ABCD}$ is a rectangle, $\text{D}$ is the center of the circle, and $\text{B}$ is on the circle. If $\text{AD}=4$ and $\text{CD}=3$ , then the area of the shaded region is between

$\text{(A)}\ 4\text{ and }5 \qquad \text{(B)}\ 5\text{ and }6 \qquad \text{(C)}\ 6\text{ and }7 \qquad \text{(D)}\ 7\text{ and...$

Solution

The area of the shaded region is equal to the area of the quarter circle with the area of the rectangle taken away. The area of the rectangle is $4\cdot 3=12$ , so we just need the quarter circle.

Applying the Pythagorean Theorem to $\triangle ADC$ , we have $(AC)^2=4^2+3^2\Rightarrow AC=5$ Since $ABCD$ is a rectangle,

Clearly $BD$ is a radius of the circle, so the area of the whole circle is $5^2\pi =25\pi$ and the area of the quarter circle is $\frac{25\pi }{4}$ .

Finally, the shaded region is $\frac{25\pi }{4}-12 \approx 7.6$ so the answer is $\boxed{\text{D}}$

1987 AJHSME (Problems • Resources)
Preceded by Problem 21	Followed by Problem 23
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

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1987 AJHSME Problems/Problem 22

From AoPSWiki

Problem

Solution

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