Question

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 80°. what area of grass will be watered?
a. \\(\frac{160}{9}pi\\) ft²
b. \\(\frac{80}{9}pi\\) ft²
c. \\(\frac{800}{9}pi\\) ft²
d. \\(\frac{200}{9}pi\\) ft²

Explanation:

Step1: Recall sector area formula

The area of a sector is $A = \frac{\theta}{360^\circ} \times \pi r^2$, where $\theta$ is the central angle and $r$ is the radius.

Step2: Identify given values

$r = 20$ ft, $\theta = 40^\circ$

Step3: Substitute values into formula

$A = \frac{40^\circ}{360^\circ} \times \pi \times (20)^2$

Step4: Simplify the expression

First, $\frac{40}{360} = \frac{1}{9}$, and $(20)^2 = 400$. So $A = \frac{1}{9} \times \pi \times 400 = \frac{800}{9}\pi \text{ ft}^2$

Answer:

C. $\frac{800}{9}\pi \text{ ft}^2$