https://file.aiquickdraw.com/mathbot/file/17276499528080wvsdpth.jpg

To solve the problem shown in your image:

We are asked to find the area of a sector-shaped field with an irrigation system, where the central angle \(\theta = 155^\circ\) and the radius \(r = 30\) yards.

### Step 1: Use the formula for the area of a sector
The area \(A\) of a sector of a circle is given by:

\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]

Where:
- \(\theta\) is the central angle in degrees,
- \(r\) is the radius of the sector.

### Step 2: Substitute the given values
\[
A = \frac{155^\circ}{360^\circ} \times \pi \times (30)^2
\]
\[
A = \frac{155}{360} \times \pi \times 900
\]
\[
A = \frac{155}{360} \times 2827.43 \quad \text{(Using} \ \pi \approx 3.1416\text{)}
\]
\[
A \approx 0.4306 \times 2827.43
\]
\[
A \approx 1218.33 \ \text{square yards}
\]

### Step 3: Round to the nearest integer
\[
A \approx 1218 \ \text{square yards}
\]

So, the area of the lawn that receives the water is approximately **1218 square yards**.

Let me know if you would like more details or if you have any further questions.

Here are some related questions to practice:
1. How does the area change if the radius is increased to 40 yards?
2. What would the area be if the angle were reduced to 120° but the radius remained the same?
3. How is the formula for the area of a sector derived?
4. If the sector were part of a circle with a different radius, say 50 yards, how would you approach the problem?
5. How can this formula be applied to real-life situations like pie-chart sectors?

**Tip:** Always make sure the angle is in degrees when using this formula unless specified otherwise.

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 = 155° and r = 30 yd.

Learn how to calculate the area of a sector-shaped field using the formula A = (θ / 360) × π × r². In this problem, θ = 155° and r = 30 yards, resulting in an area of approximately 1218 square yards. Perfect for high school students in Grades 9-12.

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