In Exercises 7–12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 1 meter Arc Length, s: 600 centimeters

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First, ensure that the units for the radius and arc length are consistent. Since the radius is given in meters and the arc length in centimeters, convert the arc length from centimeters to meters by dividing by 100: \(s = \frac{600}{100} = 6\) meters.

Recall the formula that relates the arc length \(s\), radius \(r\), and central angle \(\theta\) in radians: \(s = r \times \theta\).

Rearrange the formula to solve for the central angle \(\theta\): \(\theta = \frac{s}{r}\).

Substitute the known values of \(s = 6\) meters and \(r = 1\) meter into the formula: \(\theta = \frac{6}{1}\).

Interpret the result as the radian measure of the central angle that intercepts the given arc length on the circle.

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주요 개념

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Radian Measure of an Angle

A radian is the standard unit of angular measure, defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. It provides a direct relationship between the arc length and the radius, making it essential for measuring central angles.

Relationship Between Arc Length, Radius, and Central Angle

The central angle θ in radians is calculated using the formula θ = s / r, where s is the arc length and r is the radius. This formula links linear and angular measurements, allowing conversion from arc length to angle measure in radians.

Unit Conversion

Consistent units are crucial when applying formulas. Since the radius is given in meters and the arc length in centimeters, converting one to match the other (e.g., converting 600 cm to 6 meters) ensures accurate calculation of the central angle.