Let's solve the problem step-by-step to determine the area of grass that will be watered by the rotating sprinkler head.
Step 1: Understand the Problem
- The sprinkler sprays water up to a radius of 20 feet.
- It covers a central angle of [tex]\(80^\circ\)[/tex].
Step 2: Convert the Angle from Degrees to Radians
To use the formula for the area of a sector, we need to convert the angle from degrees to radians. The conversion factor is:
[tex]\[ 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \][/tex]
So, for [tex]\(80^\circ\)[/tex]:
[tex]\[ 80^\circ \times \frac{\pi}{180} = \frac{80\pi}{180} = \frac{4\pi}{9} \text{ radians} \][/tex]
Step 3: Use the Formula for the Area of a Sector
The formula for the area [tex]\(A\)[/tex] of a sector with radius [tex]\(r\)[/tex] and angle [tex]\(\theta\)[/tex] in radians is:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]
Here, [tex]\(r = 20 \text{ feet}\)[/tex] and [tex]\(\theta = \frac{4\pi}{9} \text{ radians} \)[/tex].
Step 4: Plug in the Values
[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]
So, the area of the grass that will be watered is:
[tex]\[ \boxed{\frac{800\pi}{9} \text{ square feet}} \][/tex]
Therefore, the correct answer is [tex]\(D. \frac{800}{9} \pi \text{ ft}^2\)[/tex].
