Let's solve for the area of the sector given that the radius of the circle is 10 feet, and the central angle of the sector is [tex]\(136^\circ\)[/tex].
### 1. Calculate the Area of the Entire Circle
First, recall the formula for the area [tex]\(A\)[/tex] of a circle with radius [tex]\(r\)[/tex]:
[tex]\[ A = \pi r^2 \][/tex]
Given the radius [tex]\(r = 10\)[/tex] feet, the area of the circle is:
[tex]\[ A = \pi \times (10)^2 = 100\pi \, \text{square feet} \][/tex]
### 2. Determine the Fraction of the Circle Represented by the Sector
The next step is to determine what fraction of the entire circle we are dealing with. The central angle is [tex]\(136^\circ\)[/tex]. Since the total degrees for any circle is [tex]\(360^\circ\)[/tex], the fraction of the circle covered by the sector is:
[tex]\[ \text{Fraction of the circle} = \frac{136}{360} \][/tex]
### 3. Calculate the Area of the Sector
Once we have the fraction of the circle that the sector represents, we multiply this fraction by the area of the entire circle:
[tex]\[ \text{Area of the sector} = \left(\frac{136}{360}\right) \times 100\pi \][/tex]
Simplifying the fraction:
[tex]\[ \frac{136}{360} = \frac{34}{90} = \frac{17}{45} \][/tex]
Therefore, the area of the sector is:
[tex]\[ \text{Area of the sector} = \left(\frac{17}{45}\right) \times 100\pi = \frac{1700\pi}{45} \][/tex]
Simplifying further:
[tex]\[ \frac{1700\pi}{45} = \frac{340\pi}{9} \, \text{square feet} \][/tex]
Hence, the area of the sector bounded by a [tex]\(136^\circ\)[/tex] arc is:
[tex]\[ \frac{340\pi}{9} \, \text{square feet} \][/tex]
### Conclusion
The correct option is:
[tex]\[
\left. \frac{340\pi}{9} \right\rvert \, \text{sq. ft.}
\][/tex]
