To find the area of a sector bounded by a given arc in a circle, we can follow these steps:
1. Calculate the radius of the circle:
The diameter of the circle is given as 8 feet. The radius ([tex]\( r \)[/tex]) is half the diameter.
[tex]\[
r = \frac{d}{2} = \frac{8}{2} = 4 \text{ feet}
\][/tex]
2. Calculate the area of the entire circle:
The area ([tex]\( A \)[/tex]) of a circle is given by the formula [tex]\( A = \pi r^2 \)[/tex].
[tex]\[
A = \pi \times (4)^2 = \pi \times 16 = 16\pi \text{ square feet}
\][/tex]
3. Determine the fraction of the circle that the sector represents:
The arc of the sector makes an angle of 36° at the center of the circle.
A full circle corresponds to 360°. The fraction of the circle's area occupied by the sector is:
[tex]\[
\text{Fraction of circle} = \frac{\text{arc degrees}}{360} = \frac{36}{360} = \frac{1}{10}
\][/tex]
4. Calculate the area of the sector:
The area of the sector is the fraction of the circle's area corresponding to that fraction.
[tex]\[
\text{Area of sector} = \text{Fraction of circle} \times \text{Area of circle} = \frac{1}{10} \times 16\pi = \frac{16\pi}{10} = \frac{8\pi}{5} \text{ square feet}
\][/tex]
Thus, the area of the sector bounded by a 36° arc in a circle with a diameter of 8 feet is:
[tex]\[
\boxed{\frac{8\pi}{5} \text{ square feet}}
\][/tex]
