Understanding the Relationship Between Arc Length, Radius, and Central Angle

The problem asks us to find the radius ($r$) of a circle given the arc length ($s$) and the central angle ($\theta$) it subtends. We are given:

• Central Angle, $\theta = 60^{\circ}$
• Arc Length, $s = 35\text{ cm}$

Applying the Arc Length Formula

The formula connecting arc length, radius, and central angle is $s = r\theta$, where $\theta$ must be in radians.

Step 1: Convert the Central Angle to Radians

To use the formula, we convert the angle from degrees to radians:

$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180^{\circ}}$

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

Step 2: Calculate the Radius

Now, substitute the known values into the arc length formula $s = r\theta$:

$35\text{ cm} = r \times \frac{\pi}{3}$

To solve for $r$, rearrange the equation:

$r = \frac{35\text{ cm}}{\frac{\pi}{3}}$

$r = 35 \times \frac{3}{\pi}\text{ cm}$

$r = \frac{105}{\pi}\text{ cm}$

Conclusion

The radius of the circle is $\frac{105}{\pi}\text{ cm}$. This corresponds to Option 1.