To determine the radian measure of a central angle corresponding to an arc in a circle, you use the formula that relates the arc length [tex]\(L\)[/tex], the radius [tex]\(r\)[/tex], and the central angle [tex]\(\theta\)[/tex] in radians:
[tex]\[ L = r \cdot \theta \][/tex]
Given:
- Radius [tex]\(r = 22\)[/tex] centimeters
- Arc length [tex]\(L = \frac{66}{5} \pi\)[/tex] centimeters
We need to find the central angle [tex]\(\theta\)[/tex]. Rearranging the formula to solve for [tex]\(\theta\)[/tex] gives us:
[tex]\[ \theta = \frac{L}{r} \][/tex]
Substituting the given values:
[tex]\[ \theta = \frac{\frac{66}{5} \pi}{22} \][/tex]
To simplify the expression:
[tex]\[ \theta = \left(\frac{66}{5} \pi \right) \div 22 \][/tex]
[tex]\[ \theta = \frac{66 \pi}{5 \cdot 22} \][/tex]
[tex]\[ \theta = \frac{66 \pi}{110} \][/tex]
[tex]\[ \theta = \frac{6 \pi}{10} \][/tex]
[tex]\[ \theta = \frac{3 \pi}{5} \][/tex]
Therefore, the radian measure of the corresponding central angle is:
[tex]\[ \theta = \frac{3}{5} \pi \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\frac{3}{5} \pi} \][/tex]
