To solve this problem, we need to determine the area of a sector of a circle. The formula for the area ([tex]\(A\)[/tex]) of a sector of a circle with radius [tex]\(r\)[/tex] and central angle [tex]\(\theta\)[/tex] in radians is:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]
Step-by-step solution:
1. Convert the angle from degrees to radians:
The angle given is [tex]\(80^\circ\)[/tex]. To convert degrees to radians, we use the following conversion factor:
[tex]\[
\theta_radians = \theta_{degrees} \times \frac{\pi}{180}
\][/tex]
Therefore,
[tex]\[
\theta_radians = 80 \times \frac{\pi}{180} = \frac{80\pi}{180} = \frac{4\pi}{9} \text{ radians}
\][/tex]
2. Plug the values into the sector area formula:
The radius ([tex]\(r\)[/tex]) is given as 20 feet, and the central angle ([tex]\(\theta\)[/tex]) in radians is [tex]\(\frac{4\pi}{9}\)[/tex].
[tex]\[
A = \frac{1}{2} \times r^2 \times \theta
\][/tex]
Substituting the given values:
[tex]\[
A = \frac{1}{2} \times 20^2 \times \frac{4\pi}{9}
\][/tex]
Simplifying inside the multiplication:
[tex]\[
A = \frac{1}{2} \times 400 \times \frac{4\pi}{9} = 200 \times \frac{4\pi}{9}
\][/tex]
[tex]\[
A = \frac{800\pi}{9}
\][/tex]
Therefore, the area of grass that will be watered by the rotating sprinkler head is [tex]\(\frac{800\pi}{9}\)[/tex] square feet.
The correct answer is:
C. [tex]\(\frac{800}{9} \pi \, \text{ft}^2\)[/tex]
