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. $\frac{800}{9}\pi\\ \text{ft}^2$
b. $\frac{200}{9}\pi\\ \text{ft}^2$
c. $\frac{80}{9}\pi\\ \text{ft}^2$
d. $\frac{760}{9}\pi\\ \text{ft}^2$

Explanation:

Step1: Recall the formula for the area of a sector

The area \( A \) of a sector of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is given by \( A=\frac{\theta}{360^{\circ}}\times\pi r^{2} \).

Step2: Identify the values of \( r \) and \( \theta \)

Here, the radius \( r = 20 \) feet (since the sprinkler sprays water as far as 20 feet) and the central angle \( \theta=80^{\circ} \).

Step3: Substitute the values into the formula

Substitute \( r = 20 \) and \( \theta = 80^{\circ} \) into the formula:
\[

$$\begin{align*} A&=\frac{80^{\circ}}{360^{\circ}}\times\pi\times(20)^{2}\\ &=\frac{80}{360}\times\pi\times400\\ &=\frac{2}{9}\times400\pi\\ &=\frac{800}{9}\pi \end{align*}$$

\]

Answer:

A. \(\frac{800}{9}\pi\ \text{ft}^2\)