Question

a water sprinkler sprays water over a distance of 27 yards while rotating through an angle of 140°. what area of lawn receives water? the area of lawn that receives the water is square yards. (round to the nearest integer as needed.)

Explanation:

Step1: Convert the angle from degrees to radians

The formula to convert degrees to radians is \(\theta_{rad}=\frac{\pi}{180}\times\theta_{deg}\).
Given \(\theta_{deg} = 140^{\circ}\), then \(\theta_{rad}=\frac{140\pi}{180}=\frac{7\pi}{9}\) radians.

Step2: Use the formula for the area of a sector

The formula for the area of a sector of a circle is \(A=\frac{1}{2}r^{2}\theta\) (where \(r\) is the radius and \(\theta\) is the central angle in radians).
Here, \(r = 27\) yards and \(\theta=\frac{7\pi}{9}\).
Substitute the values into the formula: \(A=\frac{1}{2}\times(27)^{2}\times\frac{7\pi}{9}\).

First, calculate \((27)^{2}=729\).
Then, \(\frac{1}{2}\times729\times\frac{7\pi}{9}=\frac{729\times7\pi}{18}\).
\(729\div9 = 81\), so \(\frac{729\times7\pi}{18}=\frac{81\times7\pi}{2}\).
\(81\times7 = 567\), so \(A=\frac{567\pi}{2}\approx\frac{567\times3.14159}{2}\).
\(567\times3.14159 = 1781.27153\), then \(\frac{1781.27153}{2}=890.635765\approx891\).

Answer:

\(891\)