To determine the area of grass that will be watered by the rotating sprinkler head, we need to calculate the area of the sector covered by the sprinkler.
1. Determine the radius and the central angle:
- The radius of the circle formed by the spray is [tex]\(20\)[/tex] feet.
- The central angle covered by the sprinkler is [tex]\(80^\circ\)[/tex].
2. Formula for the area of a sector:
The area of a sector of a circle with radius [tex]\(r\)[/tex] and central angle [tex]\(\theta\)[/tex] (in degrees) can be calculated using the formula:
[tex]\[
A = \frac{\theta}{360} \times \pi r^2
\][/tex]
3. Plug in the given values:
- [tex]\(r = 20\)[/tex] feet
- [tex]\(\theta = 80^\circ\)[/tex]
[tex]\[
A = \frac{80}{360} \times \pi \times 20^2
\][/tex]
4. Simplify the calculation:
First, simplify the fraction [tex]\(\frac{80}{360}\)[/tex]:
[tex]\[
\frac{80}{360} = \frac{80 \div 80}{360 \div 80} = \frac{2}{9}
\][/tex]
5. Calculate the area:
Substitute the simplified fraction and the radius into the formula:
[tex]\[
A = \frac{2}{9} \times \pi \times 20^2
\][/tex]
[tex]\[
20^2 = 400
\][/tex]
[tex]\[
A = \frac{2}{9} \times \pi \times 400
\][/tex]
[tex]\[
A = \frac{800}{9} \pi
\][/tex]
Therefore, the area of grass that will be watered by the sprinkler is:
[tex]\[
\boxed{\frac{800}{9} \pi \text{ ft}^2}
\][/tex]
So, the correct answer is:
[tex]\[ \boxed{B} \frac{800}{9} \pi \text{ ft}^2 \][/tex]
