To determine the area of grass watered by the rotating sprinkler head, we need to calculate the area of a sector of a circle. The formula for the area [tex]\( A \)[/tex] of a sector with a central angle [tex]\( \theta \)[/tex] (in radians) and radius [tex]\( r \)[/tex] is given by:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]
We are given:
- The radius [tex]\( r \)[/tex] of the circle is 20 feet.
- The central angle [tex]\( \theta \)[/tex] is [tex]\( 80^\circ \)[/tex].
Next, we need to convert the angle from degrees to radians because the formula requires the angle in radians. The conversion from degrees to radians is given by:
[tex]\[ \theta(\text{radians}) = \theta(\text{degrees}) \times \frac{\pi}{180} \][/tex]
So, converting [tex]\( 80^\circ \)[/tex] to radians:
[tex]\[ \theta(\text{radians}) = 80 \times \frac{\pi}{180} = \frac{80\pi}{180} = \frac{4\pi}{9} \][/tex]
Now, plugging these values into the area formula:
[tex]\[ A = \frac{1}{2} \times (20)^2 \times \frac{4\pi}{9} \][/tex]
[tex]\[ A = \frac{1}{2} \times 400 \times \frac{4\pi}{9} \][/tex]
[tex]\[ A = 200 \times \frac{4\pi}{9} \][/tex]
[tex]\[ A = \frac{800\pi}{9} \][/tex]
Using numerical computation (previously determined):
- The central angle in radians is approximately [tex]\( 1.396 \)[/tex] radians.
- The area watered is approximately [tex]\( 279.253 \)[/tex] square feet.
Thus, the correct area of grass that will be watered by the sprinkler is approximately [tex]\( 279.2527 \, \text{square feet} \)[/tex], corresponding to the numerical result previously determined. None of the provided options directly correspond to this specific computed area, but the step-by-step rationale, involves verifying calculations and considering any potential oversight in the provided question options.
