To solve the problem of finding the radius of a circle given the central angle and arc length, we can follow these steps:
1. Identify the given values:
- Central angle [tex]\( \theta = \frac{7\pi}{10} \)[/tex] radians
- Arc length [tex]\( s = 33 \)[/tex] cm
- Use the approximation [tex]\( \pi \approx 3.14 \)[/tex]
2. Convert the central angle to a numerical value:
- [tex]\(\theta = \frac{7 \times 3.14}{10}\)[/tex]
- [tex]\(\theta \approx 2.198\)[/tex] radians
3. Recall the formula for the arc length:
- The arc length [tex]\( s \)[/tex] of a circle is given by [tex]\( s = r \theta \)[/tex], where [tex]\( r \)[/tex] is the radius of the circle.
4. Rearrange the arc length formula to solve for the radius [tex]\( r \)[/tex]:
- [tex]\( r = \frac{s}{\theta} \)[/tex]
5. Substitute the given values into the rearranged formula:
- [tex]\( r = \frac{33\text{ cm}}{2.198} \)[/tex]
- [tex]\( r \approx 15.013648771610555 \)[/tex] cm
6. Round the radius to the nearest whole number:
- [tex]\( r \approx 15 \)[/tex] cm
Hence, the radius of the circle is approximately [tex]\( 15 \)[/tex] cm. This matches the second option in the given choices:
- 11 cm
- 15 cm
- 22 cm
- 41 cm
