To find the radius of a circle where a central angle measuring [tex]\(\frac{5 \pi}{6}\)[/tex] radians intercepts an arc of length [tex]\(35 \pi\)[/tex] centimeters, we can use the relationship between arc length ([tex]\(s\)[/tex]), radius ([tex]\(r\)[/tex]), and the central angle ([tex]\(\theta\)[/tex]) in radians. The formula for the arc length is:
[tex]\[ s = r \theta \][/tex]
Given:
- [tex]\(s = 35 \pi \)[/tex] centimeters
- [tex]\(\theta = \frac{5 \pi}{6}\)[/tex] radians
We need to isolate [tex]\(r\)[/tex] in the formula [tex]\(s = r \theta\)[/tex]. Doing this involves solving for [tex]\(r\)[/tex]:
[tex]\[ r = \frac{s}{\theta} \][/tex]
Substitute the given values for [tex]\(s\)[/tex] and [tex]\(\theta\)[/tex]:
[tex]\[ r = \frac{35 \pi}{\frac{5 \pi}{6}} \][/tex]
To simplify the expression, divide by the fraction [tex]\(\frac{5 \pi}{6}\)[/tex]:
[tex]\[ r = 35 \pi \times \frac{6}{5 \pi} \][/tex]
Notice that [tex]\(\pi\)[/tex] in the numerator and denominator cancel out:
[tex]\[ r = 35 \times \frac{6}{5} \][/tex]
Now, multiply:
[tex]\[ r = \frac{35 \times 6}{5} \][/tex]
[tex]\[ r = \frac{210}{5} \][/tex]
[tex]\[ r = 42 \][/tex]
Thus, the radius of the circle is [tex]\(42\)[/tex] centimeters.
The answer is [tex]\( \boxed{42} \)[/tex].
