To find the length of the minor arc [tex]\( SV \)[/tex] in a circle with radius [tex]\( r = 24 \)[/tex] inches and central angle [tex]\( \theta = \frac{5 \pi}{6} \)[/tex] radians, we can use the formula for the arc length:
[tex]\[ S = r \cdot \theta \][/tex]
Where:
- [tex]\( r \)[/tex] is the radius of the circle.
- [tex]\( \theta \)[/tex] is the central angle in radians.
Given:
- [tex]\( r = 24 \)[/tex] inches
- [tex]\( \theta = \frac{5 \pi}{6} \)[/tex]
First, substitute the given values into the arc length formula:
[tex]\[ S = 24 \cdot \frac{5 \pi}{6} \][/tex]
Next, perform the multiplication:
[tex]\[ S = 24 \cdot \frac{5 \pi}{6} \][/tex]
[tex]\[ S = 24 \cdot \frac{5 \pi}{6} = 24 \cdot \frac{5 \pi}{6} = 24 \cdot \frac{5}{6} \cdot \pi \][/tex]
[tex]\[ S = (24 \cdot \frac{5}{6}) \cdot \pi = (24 \cdot \frac{5}{6}) \cdot \pi = (24 \cdot \frac{5}{6}) \cdot \pi = (24 \cdot \frac{5}{6}) \cdot \pi = 24 \cdot \frac{5}{6} \][/tex]
[tex]\[ S = 24 \cdot \frac{5}{6} = \frac{120}{6} = 20 \][/tex]
[tex]\[ S = 20 \pi \][/tex]
Thus, the length of the minor arc [tex]\( SV \)[/tex] is [tex]\( 20 \pi \)[/tex] inches.
