To determine the length of the circle's radius given the arc length and the central angle, we can use the relationship between arc length, radius, and central angle in radians. Let's go through the steps to solve this:
1. Identify the given values:
- Arc length, [tex]\( s = \frac{20}{3} \pi \)[/tex] centimeters
- Central angle, [tex]\( \theta = 65^\circ \)[/tex]
2. Convert the central angle from degrees to radians:
- The formula to convert degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
- Substituting [tex]\( 65^\circ \)[/tex]:
[tex]\[
\theta_{\text{rad}} = 65 \times \frac{\pi}{180} \approx 1.134464 \, \text{radians}
\][/tex]
3. Use the formula for arc length:
- The formula for arc length is given by:
[tex]\[
s = r \theta
\][/tex]
- Rearranging this formula to solve for the radius [tex]\( r \)[/tex]:
[tex]\[
r = \frac{s}{\theta}
\][/tex]
4. Substitute the given values into the formula:
- Using [tex]\( s = \frac{20}{3} \pi \)[/tex] and [tex]\( \theta_{\text{rad}} \approx 1.134464 \)[/tex]:
[tex]\[
r = \frac{\frac{20}{3} \pi}{1.134464}
\][/tex]
5. Calculate the value of the radius:
[tex]\[
r \approx \frac{20.943951023931955}{1.1344640137963142} \approx 18.461538461538463 \, \text{centimeters}
\][/tex]
Thus, the length of the circle's radius is approximately [tex]\( 18.46 \)[/tex] centimeters, which is not one of the options provided (2 cm, 8 cm, 16 cm, 4 cm). Therefore, with the calculated value, none of the given options are correct based on the provided conditions.
