Find the length to three significant digits of each arc intercepted by a central angle in a circle of radius r. See Example 1. r = 15.1 in. , θ = 210°

Verified step by step guidance

1. Convert the angle from degrees to radians using the formula: \( \text{radians} = \theta \times \frac{\pi}{180} \).
2. Substitute \( \theta = 210^\circ \) into the formula to find the angle in radians.
3. Use the formula for the arc length \( s = r \times \theta \) where \( r \) is the radius and \( \theta \) is the angle in radians.
4. Substitute \( r = 15.1 \) inches and the calculated \( \theta \) in radians into the arc length formula.
5. Calculate the arc length to three significant digits.

Key Concepts

Arc Length Formula

The arc length of a circle can be calculated using the formula L = rθ, where L is the arc length, r is the radius, and θ is the central angle in radians. To convert degrees to radians, use the conversion factor π/180. This formula is essential for determining the length of the arc intercepted by a given central angle.

Central Angle

A central angle is formed by two radii of a circle that meet at the center. The measure of the central angle directly influences the length of the arc it intercepts. Understanding how to measure and convert this angle from degrees to radians is crucial for applying the arc length formula correctly.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. When reporting the length of the arc, it is important to round the result to three significant digits to ensure clarity and accuracy in communication. This practice is common in scientific and engineering contexts to reflect the precision of measurements.