To find the length of a minor arc in a circle, we use the formula for the arc length:
[tex]\[
\text{Arc length} = r \cdot \theta
\][/tex]
where [tex]\( r \)[/tex] is the radius of the circle, and [tex]\( \theta \)[/tex] is the central angle in radians.
Given:
- The radius [tex]\( r \)[/tex] of circle [tex]\( T \)[/tex] is [tex]\( 24 \)[/tex] inches.
- The central angle [tex]\( \theta \)[/tex] is [tex]\( \frac{5 \pi}{6} \)[/tex] radians.
Substituting the values into the arc length formula:
[tex]\[
\text{Arc length} = 24 \cdot \left( \frac{5 \pi}{6} \right)
\][/tex]
First, perform the multiplication inside the parentheses:
[tex]\[
24 \cdot \frac{5 \pi}{6} = 24 \cdot \frac{5 \pi}{6} = 24 \cdot \frac{5}{6} \cdot \pi
\][/tex]
Simplify [tex]\( 24 \cdot \frac{5}{6} \)[/tex]:
[tex]\[
24 \div 6 = 4
\][/tex]
Then:
[tex]\[
4 \cdot 5 = 20
\][/tex]
Thus:
[tex]\[
24 \cdot \frac{5 \pi}{6} = 20 \pi
\][/tex]
Therefore, the length of the minor arc [tex]\( SV \)[/tex] is:
[tex]\[
20 \pi \, \text{in.}
\][/tex]
So, the correct answer is:
[tex]\[
20 \pi \, \text{in.}
\][/tex]
