- Identify the radius \(r = 30\) and the central angle \(\theta = \frac{\pi}{6}\).
- Use the arc length formula \(s = r \cdot \theta\) to compute the exact arc length as \(5\pi\).
- Approximate the value using \(\pi \approx 3.14159\) to get \(15.70795\).
- Round the final result to the nearest tenth to get \(\boxed{15.7 \text{ cm}}\).
Explanation
- Define the arc length formula and identify given values.
The problem asks for the arc length of a circle \(N\) with a radius \(r = 30 \text{ cm}\) subtended by a central angle \(\theta = \frac{\pi}{6} \text{ radians}\). The arc length \(s\) of a circle is defined by the formula \(s = r \cdot \theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.
- Calculate the exact and numerical value of the arc length.
Using the provided values, we substitute \(r = 30 \text{ cm}\) and \(\theta = \frac{\pi}{6} \text{ radians}\) into the arc length formula:
\[s = 30 \times \frac{\pi}{6}\]
Simplifying this expression, we get:
\[s = 5\pi \text{ cm}\]
Using the value of \(\pi \approx 3.14159\), we calculate:
\[s = 5 \times 3.14159 \approx 15.70795 \text{ cm}\]
- Round to the nearest tenth and identify the correct option.
Rounding the calculated result \(15.70795\) to the nearest tenth gives \(15.7 \text{ cm}\). Comparing this to the given options, we find that the correct length of the minor arc \(LM\) is \(15.7 \text{ centimeters}\).
Examples
Arc length is a fundamental concept used in various fields like civil engineering and design. For instance, when designing a curved road, engineers must calculate the arc length of the circular segment to determine the exact amount of asphalt needed for the road surface. By knowing the radius of the turn and the angle of the curve, they can precisely calculate the length and ensure a smooth, cost-effective construction.\[s = r \cdot \theta\]where \(r\) is the radius of the turn and \(\theta\) is the central angle.