In the figure, \( \overline{AO} = 4 \). What is the area of the shaded region?
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\( \pi \)
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\( 3\pi \)
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\( 2\pi \)
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\( 6\pi \)
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\( 4\pi \)
Try to solve it before checking the answer.
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\( 4\pi \)
Indeed; let be:
- \( A_{so} \): area of the shaded region.
- \(r\): radius of the semicircle.
- \(r’\): radius of the inner circle.
- \( A_{sc} \): area of the semicircle.
- \( A_{ci} \): area of the inner circle.
We have,
\[ r= 4 \quad \text{ y } r’ = 2 \]Moreover,
\[ \begin{aligned} A_{so} &= A_{sc} – A_{ci} \\[2em] &= \frac{\pi r^2}{2} – \pi r’^2 \\[2em] &= \frac{4^2 \pi}{2} – 2^2 \pi \\[2em] &= 8\pi – 4\pi = \boldsymbol{4\pi} \end{aligned} \]