Work each problem. See Example 5. Irrigation Area 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 = 40.0° and r = 152 yd.

Verified step by step guidance

1. Identify the shape of the field as a sector of a circle, where the radius is given as \(r = 152\) yards and the central angle is \(\theta = 40.0^\circ\).
2. Recall the formula for the area of a sector of a circle: \(\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2\), where \(\theta\) is in degrees.
3. Substitute the given values into the formula: replace \(\theta\) with 40.0 and \(r\) with 152.
4. Calculate the area by first squaring the radius (\(r^2\)), then multiplying by \(\pi\), and finally multiplying by the fraction \(\frac{\theta}{360^\circ}\).
5. Express the final answer in square yards, which represents the area of the sector-shaped field irrigated by the system.

Key Concepts

Sector Area Formula

The area of a sector of a circle is given by (θ/360) × π × r², where θ is the central angle in degrees and r is the radius. This formula calculates the portion of the circle's area corresponding to the given angle, essential for finding the field's area shaped like a sector.

Central Angle in Degrees

The central angle θ defines the size of the sector and is measured in degrees. Understanding how to use this angle in the sector area formula is crucial, as it determines the fraction of the full circle's area that the sector occupies.

Radius of the Sector

The radius r is the distance from the center of the circle to the boundary of the sector. It represents the length of the irrigation system's reach and is squared in the area formula, significantly impacting the total area calculation.