To determine the area of grass that will be watered, we need to calculate the area of a sector of a circle. The formula for the area of a sector is given by:
[tex]\[ \text{Area of sector} = \left(\frac{\theta}{360}\right) \times \pi \times r^2 \][/tex]
where:
- [tex]\(\theta\)[/tex] is the central angle in degrees,
- [tex]\(r\)[/tex] is the radius of the circle.
Given:
- Radius [tex]\(r = 20\)[/tex] feet,
- Central angle [tex]\(\theta = 80^\circ\)[/tex].
Let's substitute these values into the formula.
1. Plug in the values:
[tex]\[
\text{Area of sector} = \left(\frac{80}{360}\right) \times \pi \times (20^2)
\][/tex]
2. Simplify the fraction:
[tex]\[
\frac{80}{360} = \frac{2}{9}
\][/tex]
3. Substitute this into the formula:
[tex]\[
\text{Area of sector} = \left(\frac{2}{9}\right) \times \pi \times 400
\][/tex]
4. Perform the multiplication:
[tex]\[
\left(\frac{2}{9}\right) \times 400 = \frac{800}{9}
\][/tex]
5. Combine the values with [tex]\(\pi\)[/tex]:
[tex]\[
\text{Area of sector} = \frac{800}{9} \pi \, \text{ft}^2
\][/tex]
Thus, the area of grass that will be watered is:
[tex]\[\boxed{\frac{800}{9} \pi \, \text{ft}^2}\][/tex]
Therefore, the correct answer is:
- C. [tex]\(\frac{800}{9} \pi \, \text{ft}^2\)[/tex]
