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 = 145° and r = 36 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 formula for the area \( A \) of a sector with radius \( r \) and central angle \( \theta \) (in degrees) is \( A=\frac{\theta}{360^{\circ}}\times\pi r^{2} \).

Step2: Substitute the given values

We are given \( \theta = 145^{\circ} \) and \( r = 36 \) yd. Substitute these values into the formula:
\( A=\frac{145^{\circ}}{360^{\circ}}\times\pi\times(36)^{2} \)

Step3: Simplify the expression

First, calculate \( (36)^{2}=1296 \). Then, \( \frac{145}{360}\times\pi\times1296 \). Simplify \( \frac{145}{360}\times1296 \):
\( \frac{145\times1296}{360}=\frac{145\times1296\div 72}{360\div 72}=\frac{145\times18}{5}=145\times\frac{18}{5}=145\times3.6 = 522 \)
So, \( A = 522\pi \)

Step4: Calculate the numerical value

Now, calculate \( 522\pi\approx522\times3.1416\approx1639.9152 \)

Step5: Round to the nearest integer

Rounding \( 1639.9152 \) to the nearest integer gives \( 1640 \).

Answer:

\( 1640 \)