To determine the area of the grass that will be watered by the rotating sprinkler head, we will calculate the area of a circular sector.
The key pieces of information we are given are:
- The radius [tex]\( r \)[/tex] of the sector, which is 20 feet.
- The central angle [tex]\( \theta \)[/tex], which is [tex]\( 80^\circ \)[/tex].
To find the area of a sector of a circle, we use the formula:
[tex]\[ \text{Area of sector} = \left(\frac{\theta}{360}\right) \times \pi r^2 \][/tex]
Here’s the step-by-step calculation:
1. Identify the radius and central angle:
[tex]\[ r = 20 \text{ feet}, \quad \theta = 80^\circ \][/tex]
2. Substitute the values into the formula:
[tex]\[ \text{Area of sector} = \left(\frac{80}{360}\right) \times \pi \times (20)^2 \][/tex]
3. Simplify the fraction:
[tex]\[ \frac{80}{360} = \frac{2}{9} \][/tex]
4. Calculate the squared radius:
[tex]\[ 20^2 = 400 \][/tex]
5. Multiply and simplify:
[tex]\[ \left(\frac{2}{9}\right) \times \pi \times 400 = \frac{800}{9} \pi \text{ square feet} \][/tex]
Hence, the correct answer is:
[tex]\[ \frac{800}{9} \pi \text{ square feet} \][/tex]
Comparing with the provided options:
A. [tex]$\frac{8}{8} \pi ft ^2$[/tex]
B. [tex]$\frac{T 00}{2} \pi ft ^2$[/tex]
C. [tex]$\frac{80}{2} \pi ft ^2$[/tex]
None of the provided options exactly match our simplified form. However, we realize there may be a typographical error in the options or in the original list of choices. The numerical result from our calculation, [tex]\( \frac{800}{9} \pi \approx 279.2526803190927 \text{ square feet} \)[/tex], confirms the correct area as calculated.
