To determine the area of grass watered by the sprinkler, we need to calculate the area of the sector formed by the central angle [tex]\(80^\circ\)[/tex] with a radius of 20 feet. A sector of a circle is a portion of the circle that resembles a 'slice of pie' and its area can be calculated using the following steps:
1. Convert the Angle to Radians:
The given angle is [tex]\(80^\circ\)[/tex]. To use it in calculations, we need to convert it to radians.
[tex]\[
\text{angle in radians} = 80^\circ \times \frac{\pi}{180^\circ} = \frac{80\pi}{180} = \frac{4\pi}{9} \approx 1.396
\][/tex]
2. Calculate the Area of the Full Circle:
The area [tex]\(A\)[/tex] of a full circle with radius [tex]\(r = 20\)[/tex] feet is given by the formula:
[tex]\[
A_{\text{circle}} = \pi r^2 = \pi (20)^2 = 400\pi
\][/tex]
3. Calculate the Area of the Sector:
The area of the sector (portion of the circle covered by the sprinkler) can be determined by the ratio of the central angle to the full circle angle [tex]\(2\pi\)[/tex] radians.
[tex]\[
\text{Area of sector} = \left( \frac{\theta}{2\pi} \right) \times A_{\text{circle}}
\][/tex]
Here, [tex]\(\theta = \frac{4\pi}{9}\)[/tex] is the central angle in radians.
[tex]\[
\text{Area of sector} = \left( \frac{\frac{4\pi}{9}}{2\pi} \right) \times 400\pi = \left( \frac{4}{18} \right) \times 400\pi = \frac{2}{9} \times 400\pi = \frac{800}{9} \pi \text{ square feet}
\][/tex]
Thus, the correct area of the grass watered by the sprinkler is [tex]\(\frac{800}{9} \pi\)[/tex] square feet. Therefore, the correct answer is:
C. [tex]\(\frac{800}{9} \pi \, \text{ft}^2\)[/tex]
