To determine the area of grass watered by the sprinkler, we need to calculate the area of a sector of a circle. The sector is defined by a radius (r) and a central angle (θ).
Given:
- Radius ([tex]\(r\)[/tex]) = 20 feet
- Central angle ([tex]\(\theta\)[/tex]) = [tex]\(80^\circ\)[/tex]
The formula for the area of a sector of a circle is given by:
[tex]\[
A = \frac{1}{2} r^2 \theta \text{ (in radians)}
\][/tex]
First, we need to convert the angle from degrees to radians because the formula for the sector's area requires the angle in radians. We use the conversion factor:
[tex]\[
1 \text{ degree} = \frac{\pi}{180} \text{ radians}
\][/tex]
So, converting [tex]\(80^\circ\)[/tex] to radians:
[tex]\[
\theta \text{ (in radians)} = 80^\circ \times \frac{\pi}{180}
\][/tex]
Simplifying the conversion:
[tex]\[
\theta \text{ (in radians)} = \frac{80\pi}{180} = \frac{4\pi}{9}
\][/tex]
Now, we can substitute the radius and the angle in radians into the formula for the area of the sector:
[tex]\[
A = \frac{1}{2} \times 20^2 \times \frac{4\pi}{9}
\][/tex]
Calculating the constants first:
[tex]\[
20^2 = 400
\][/tex]
Then:
[tex]\[
A = \frac{1}{2} \times 400 \times \frac{4\pi}{9} = 200 \times \frac{4\pi}{9} = \frac{800\pi}{9}
\][/tex]
So, the area of the grass that will be watered is:
[tex]\[
A = \frac{800}{9} \pi \, \text{ft}^2
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{800}{9} \pi \, \text{ft}^2}
\][/tex]
Therefore, the correct option is:
D. [tex]\(\frac{800}{9} \pi \, \text{ft}^2\)[/tex]
