Answer :
To find the length of the minor arc SV in a circle with radius [tex]\( r = 24 \)[/tex] inches and a central angle [tex]\( \theta = \frac{5\pi}{6} \)[/tex] radians, we can use the formula for the length of an arc in a circle:
[tex]\[ L = r \cdot \theta \][/tex]
### Step-by-Step Solution:
1. Identify the given parameters:
- Radius [tex]\( r = 24 \)[/tex] inches.
- Central angle [tex]\( \theta = \frac{5\pi}{6} \)[/tex] radians.
2. Substitute the given values into the formula:
[tex]\[ L = 24 \text{ inches} \times \frac{5\pi}{6} \][/tex]
3. Perform the multiplication:
[tex]\[ L = 24 \times \frac{5\pi}{6} \][/tex]
4. First, simplify the fraction:
[tex]\[ \frac{5\pi}{6} \][/tex]
5. Multiply the radius by the simplified fraction:
[tex]\[ L = 24 \times \frac{5\pi}{6} \][/tex]
6. Perform the multiplication:
The factor of 24 divided by 6 gives:
[tex]\[ L = 4 \times 5\pi = 20\pi \][/tex]
Thus, the length of minor arc SV is:
[tex]\[ L = 20\pi \text{ inches} \][/tex]
From the given options, the correct answer is:
[tex]\[ \boxed{20 \pi} \text{ inches} \][/tex]
[tex]\[ L = r \cdot \theta \][/tex]
### Step-by-Step Solution:
1. Identify the given parameters:
- Radius [tex]\( r = 24 \)[/tex] inches.
- Central angle [tex]\( \theta = \frac{5\pi}{6} \)[/tex] radians.
2. Substitute the given values into the formula:
[tex]\[ L = 24 \text{ inches} \times \frac{5\pi}{6} \][/tex]
3. Perform the multiplication:
[tex]\[ L = 24 \times \frac{5\pi}{6} \][/tex]
4. First, simplify the fraction:
[tex]\[ \frac{5\pi}{6} \][/tex]
5. Multiply the radius by the simplified fraction:
[tex]\[ L = 24 \times \frac{5\pi}{6} \][/tex]
6. Perform the multiplication:
The factor of 24 divided by 6 gives:
[tex]\[ L = 4 \times 5\pi = 20\pi \][/tex]
Thus, the length of minor arc SV is:
[tex]\[ L = 20\pi \text{ inches} \][/tex]
From the given options, the correct answer is:
[tex]\[ \boxed{20 \pi} \text{ inches} \][/tex]
