Sure, let's solve this problem step-by-step.
1. Understanding the problem:
- We have a circle with a central angle of [tex]\(\frac{7\pi}{10}\)[/tex] radians.
- This angle intercepts an arc with a length of 33 cm.
- We need to find the radius of this circle and round it to the nearest whole centimeter.
- We will use [tex]\(3.14\)[/tex] as the value for [tex]\(\pi\)[/tex].
2. Formula:
- The formula for the length of an arc [tex]\( L \)[/tex] in a circle is given by:
[tex]\[
L = r \cdot \theta
\][/tex]
where [tex]\( L \)[/tex] is the arc length, [tex]\( r \)[/tex] is the radius of the circle, and [tex]\( \theta \)[/tex] is the central angle in radians.
3. Substitute the known values:
- We know the arc length [tex]\( L = 33 \)[/tex] cm.
- The central angle [tex]\( \theta = \frac{7\pi}{10} \)[/tex]:
[tex]\[
\theta = \frac{7 \times 3.14}{10}
\][/tex]
[tex]\[
\theta = 2.198
\][/tex]
4. Solve for the radius [tex]\( r \)[/tex]:
- Rearrange the arc length formula to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{L}{\theta}
\][/tex]
- Substituting the values:
[tex]\[
r = \frac{33}{2.198}
\][/tex]
[tex]\[
r \approx 15.013648771610555
\][/tex]
5. Rounding the radius to the nearest whole cm:
- The result is approximately [tex]\( 15.013648771610555 \)[/tex].
- Rounding [tex]\( 15.013648771610555 \)[/tex] to the nearest whole number gives us [tex]\( 15 \)[/tex] cm.
Therefore, the radius of the circle is approximately [tex]\( 15 \)[/tex] cm when rounded to the nearest whole centimeter. The correct answer is:
15 cm
