$\"\"$

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A security light mounted above a house illuminates part of a circular region defined by

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$\"\"$ and $\"\"$ .

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The area of a sector of a circle with radius $\"\"$ and central angle $\"\"$ is $\"\"$ .

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Where $\"\"$ is measured in radians.

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The total area of the region is approximately $\"\"$ .

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The area of the region described is the difference of the areas of two sectors.

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Both sectors have an angle measure of $\"\"$ .

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By solving $\"\"$ ,we get $\"\"$ .

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The larger radius is $\"\"$ and the smaller radius is $\"\"$ .

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$\"\"$

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Calculate the $\"\"$ by substituting the values of $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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The total area of the region is approximately $\"\"$ .

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Substitute this value in for $\"\"$ in the equation above and solve for $\"\"$ .

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$\"\"$

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Since $\"\"$ is not in negative, it is not considered.

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Therefore, the value of $\"\"$ is $\"\"$ .

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$\"\"$

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The value of $\"\"$ is $\"\"$ .