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°

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Identify the given values: the radius of the circle \(r = 15.1\) inches and the central angle \(\theta = 210^\circ\).

Recall the formula for the length of an arc intercepted by a central angle in a circle: \(\text{Arc length} = r \times \theta_{\text{radians}}\).

Convert the central angle from degrees to radians using the conversion factor \(\pi\) radians = \(180^\circ\). The formula is \(\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\).

Substitute the given angle into the conversion formula to find \(\theta\) in radians: \(210^\circ \times \frac{\pi}{180}\).

Multiply the radius \(r\) by the angle in radians to find the arc length: \(15.1 \times \theta_{\text{radians}}\). This will give the arc length in inches.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Arc Length Formula

The arc length of a circle segment is calculated using the formula s = rθ, where r is the radius and θ is the central angle in radians. This formula relates the linear distance along the circle's edge to the angle subtended at the center.

Conversion Between Degrees and Radians

Since the arc length formula requires the angle in radians, converting degrees to radians is essential. Use the conversion factor π radians = 180°, so θ (radians) = θ (degrees) × π / 180.

Significant Figures in Measurement

When reporting the arc length, it is important to round the result to the specified number of significant digits, here three. This ensures the precision of the answer matches the given data's accuracy.

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