A circle has a radius of 30 cm. Determine the length of the arc subtended by an angle of

$60^\circ$

at the center of the circle.

<h2>Question:</h2>

A circle has a radius of 30 cm. Determine the length of the arc subtended by an angle of $60^\circ$ at the center of the circle.

<h2>Choices:</h2>

• $(a) \frac{\pi}{6} \text{ cm}$

• $(b) 20\pi \text{ cm}$

• $(c) 10\pi \text{ cm}$

• $(d) 6\pi \text{ cm}$

<h2>Correct Answer:</h2>

$(c) 10\pi \text{ cm}$

<h2>Explanation:</h2>

To find the length of the arc subtended by an angle at the center of the circle, we use the formula:

Arc Length Formula: $L = \theta \cdot r$

where:

• $L$ is the length of the arc.

• $\theta$ is the central angle in radians.

• $r$ is the radius of the circle.

First, we need to convert the central angle from degrees to radians. The conversion factor between degrees and radians is $\frac{\pi}{180}$:

$$\theta = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}$$

Now that we have the angle in radians, we can use the Arc Length Formula. The radius $r$ is given as 30 cm:

$$L = \theta \cdot r = \frac{\pi}{3} \cdot 30 \text{ cm}$$

$$L = 10\pi \text{ cm}$$

Therefore, the length of the arc subtended by an angle of $60^\circ$ at the center of the circle is $10\pi$ cm, making option $(c)$ the correct answer.

<h2>Conclusion:</h2>

This explanation demonstrates how to convert the angle to radians and then use the formula for the arc length. Applying these steps ensures we arrive at the correct answer, $10\pi$ cm.