Select the correct answer.

A rotating sprinkler head sprays water as far as 20 feet. The head is set to cover a central angle of [tex]$80^{\circ}$[/tex]. What area of grass will be watered?

A. [tex]\frac{80}{9} \pi \, \text{ft}^2[/tex]
B. [tex]\frac{200}{9} \pi \, \text{ft}^2[/tex]
C. [tex]\frac{800}{9} \pi \, \text{ft}^2[/tex]
D. [tex]\frac{760}{9} \pi \, \text{ft}^2[/tex]



Answer :

To determine the area of grass that will be watered, we need to calculate the area of a sector of a circle. The formula for the area of a sector is given by:

[tex]\[ \text{Area of sector} = \left(\frac{\theta}{360}\right) \times \pi \times r^2 \][/tex]

where:
- [tex]\(\theta\)[/tex] is the central angle in degrees,
- [tex]\(r\)[/tex] is the radius of the circle.

Given:
- Radius [tex]\(r = 20\)[/tex] feet,
- Central angle [tex]\(\theta = 80^\circ\)[/tex].

Let's substitute these values into the formula.

1. Plug in the values:
[tex]\[ \text{Area of sector} = \left(\frac{80}{360}\right) \times \pi \times (20^2) \][/tex]

2. Simplify the fraction:
[tex]\[ \frac{80}{360} = \frac{2}{9} \][/tex]

3. Substitute this into the formula:
[tex]\[ \text{Area of sector} = \left(\frac{2}{9}\right) \times \pi \times 400 \][/tex]

4. Perform the multiplication:
[tex]\[ \left(\frac{2}{9}\right) \times 400 = \frac{800}{9} \][/tex]

5. Combine the values with [tex]\(\pi\)[/tex]:
[tex]\[ \text{Area of sector} = \frac{800}{9} \pi \, \text{ft}^2 \][/tex]

Thus, the area of grass that will be watered is:

[tex]\[\boxed{\frac{800}{9} \pi \, \text{ft}^2}\][/tex]

Therefore, the correct answer is:
- C. [tex]\(\frac{800}{9} \pi \, \text{ft}^2\)[/tex]