To determine the area of grass that will be watered by the sprinkler, we need to find the area of a sector of a circle. A sector is essentially a slice of the circle, defined by the radius and the central angle.
Here, we are told:
- The radius of the circle is 20 feet.
- The central angle is [tex]\(80^\circ\)[/tex].
First, we need to convert the central angle from degrees to radians because the formula to find the area of a sector typically uses radians. The conversion factor from degrees to radians is:
[tex]\[
1 \text{ degree} = \frac{\pi \text{ radians}}{180^\circ}
\][/tex]
So, converting [tex]\(80^\circ\)[/tex] to radians:
[tex]\[
80^\circ \times \frac{\pi \text{ radians}}{180^\circ} = \frac{80\pi}{180} = \frac{4\pi}{9}
\][/tex]
Now we have the central angle in radians as [tex]\(\frac{4\pi}{9}\)[/tex].
Next, we use the formula for the area of a sector, which is:
[tex]\[
\text{Area of a sector} = \frac{\text{central angle in radians}}{2\pi} \times \pi \times \text{radius}^2
\][/tex]
Substituting the values we have:
[tex]\[
\text{Area} = \frac{\frac{4\pi}{9}}{2\pi} \times \pi \times 20^2
\][/tex]
Simplify inside the formula:
[tex]\[
\text{Area} = \left(\frac{4\pi}{9} \times \frac{1}{2\pi}\right) \times \pi \times 400
\][/tex]
[tex]\[
\text{Area} = \left(\frac{4}{18}\right) \times 400
\][/tex]
[tex]\[
\text{Area} = \left(\frac{2}{9}\right) \times 400
\][/tex]
[tex]\[
\text{Area} = \frac{800}{9}
\][/tex]
So, the area of the sector, or the area of grass that will be watered, is:
[tex]\[
\frac{800}{9} \pi \ \text{square feet}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{800}{9} \pi \ \text{ft}^2}
\][/tex]
The correct choice is option B.
