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 = 12.3 cm , θ = 2π/3 radians

Verified step by step guidance

1. Identify the formula for the arc length, which is given by \( s = r \theta \), where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians.
2. Substitute the given values into the formula: \( r = 12.3 \text{ cm} \) and \( \theta = \frac{2\pi}{3} \text{ radians} \).
3. Calculate the product of the radius and the angle: \( s = 12.3 \times \frac{2\pi}{3} \).
4. Simplify the expression by multiplying the numbers and keeping \( \pi \) as part of the expression.
5. Round the result to three significant digits to find the arc length.

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. This formula derives from the relationship between the angle and the proportion of the circle's circumference that the arc represents.

Central Angle

A central angle is an angle whose vertex is at the center of the circle and whose sides extend to the circumference. The measure of the central angle in radians directly relates to the length of the arc it intercepts, making it crucial for calculating arc lengths.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. When reporting measurements, such as the length of an arc, it is important to express the result to the correct number of significant digits, which reflects the precision of the given data.