Question

a center-pivot irrigation system provides water to a sector-shaped field as shown in the figure. find the area of the field if θ = 125° and r = 30 yd. the area of lawn that receives the water is square yards. (round to the nearest integer as needed.)

Explanation:

Step1: Recall the formula for the area of a sector

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

Step2: Substitute the given values

We are given \( \theta = 125^{\circ} \) and \( r = 30 \) yd. Substitute these values into the formula:
\( A=\frac{125}{360}\times\pi\times(30)^{2} \)

Step3: Simplify the expression

First, calculate \( (30)^{2}=900 \). Then,
\( A=\frac{125}{360}\times\pi\times900 \)
Simplify \( \frac{125\times900}{360} \):
\( \frac{125\times900}{360}=\frac{125\times5}{2}=\frac{625}{2} = 312.5 \)
So, \( A = 312.5\times\pi \)

Step4: Calculate the numerical value

Using \( \pi\approx3.1416 \), we get:
\( A\approx312.5\times3.1416 = 981.75 \)
Rounding to the nearest integer, we get \( 982 \).

Answer:

982