Find the area of a sector of a circle having radius r and central angle θ. Express answers to the nearest tenth. See Example 5. r = 30.0 ft, θ = π/2 radians

Verified step by step guidance

1. Identify the formula for the area of a sector: \( A = \frac{1}{2} r^2 \theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians.
2. Substitute the given values into the formula: \( r = 30.0 \) ft and \( \theta = \frac{\pi}{2} \) radians.
3. Calculate \( r^2 \) by squaring the radius: \( 30.0^2 \).
4. Multiply the result of \( r^2 \) by \( \theta \) and then by \( \frac{1}{2} \) to find the area.
5. Round the final result to the nearest tenth to express the area of the sector.

Key Concepts

Area of a Sector

The area of a sector of a circle is a portion of the circle defined by a central angle. It can be calculated using the formula A = (1/2) * r^2 * θ, where A is the area, r is the radius, and θ is the central angle in radians. This formula derives from the relationship between the angle and the total area of the circle.

Radians

Radians are a unit of angular measure used in mathematics. One radian is the angle formed when the arc length is equal to the radius of the circle. This unit is essential in trigonometry as it simplifies the calculations involving circular functions and is often preferred over degrees in calculus and physics.

Units of Measurement

In solving problems involving geometry and trigonometry, it is crucial to maintain consistent units of measurement. In this case, the radius is given in feet, and the area will be expressed in square feet. Understanding how to convert and express measurements accurately is vital for obtaining correct results.