Step-by-Step Solution:
To find the length of the minor arc [tex]\(SV\)[/tex] in circle [tex]\(T\)[/tex] with a radius of [tex]\(24\)[/tex] inches and an angle [tex]\(\theta = \frac{5\pi}{6}\)[/tex] radians, follow these steps:
1. Understand the problem:
- Radius of the circle, [tex]\(r = 24\)[/tex] inches.
- Central angle in radians, [tex]\(\theta = \frac{5\pi}{6}\)[/tex].
2. Recall the formula for the arc length:
The formula for the arc length [tex]\(L\)[/tex] of a circle is given by:
[tex]\[
L = r \cdot \theta
\][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is the central angle in radians.
3. Substitute the given values into the formula:
- [tex]\(r = 24\)[/tex] inches.
- [tex]\(\theta = \frac{5\pi}{6}\)[/tex] radians.
So,
[tex]\[
L = 24 \cdot \frac{5\pi}{6}
\][/tex]
4. Perform the multiplication:
First, multiply the constants:
[tex]\[
24 \cdot \frac{5}{6} = 24 \cdot \frac{5}{6} = 4 \cdot 5 = 20
\][/tex]
Therefore,
[tex]\[
L = 20 \cdot \pi = 20\pi
\][/tex]
Conclusion:
The length of the minor arc [tex]\(SV\)[/tex] is [tex]\(20\pi\)[/tex] inches.
Answer:
[tex]\[ 20\pi \, \text{in.} \][/tex]
