To find the area of a sector of a circle, we can use the formula:
\[ \text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
where \(\theta\) is the angle of the sector in degrees, and \(r\) is the radius of the circle.
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Find the radius: The diameter of the circle is given as 18 inches, so the radius \(r\) is:
\[ r = \frac{\text{diameter}}{2} = \frac{18}{2} = 9 \text{ inches} \]
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Insert the values into the formula: We have \(\theta = 122^\circ\) and \(r = 9\) inches.
Now, plugging these values into the sector area formula:
\[ \text{Area of sector} = \frac{122}{360} \times \pi \times 9^2 \]
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Calculate \(9^2\):
\[ 9^2 = 81 \]
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Now calculate the area:
\[ \text{Area of sector} = \frac{122}{360} \times \pi \times 81 \]
\[ \text{Area of sector} = \frac{122 \times 81}{360} \times \pi \]
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Calculate \(\frac{122 \times 81}{360}\):
\[ 122 \times 81 = 9870 \]
Now divide by 360:
\[ \frac{9870}{360} \approx 27.41667 \]
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Now multiply by \(\pi\):
Using \(\pi \approx 3.14159\):
\[ \text{Area of sector} \approx 27.41667 \times 3.14159 \approx 86.1757 \]
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Final Step: Round to two decimal places:
\[ \text{Area of sector} \approx 86.18 \text{ square inches} \]
Thus, the area of the sector is approximately 86.18 square inches.