Answer :
To find the length of the intercepted arc when given a central angle and the radius of the circle, you can use the following formula for arc length:
[tex]\[ \text{Arc Length} = \text{Radius} \times \text{Central Angle (in radians)} \][/tex]
Here are the steps to find the arc length:
1. Identify the given values:
- Central angle: [tex]\(\frac{5\pi}{6}\)[/tex] radians
- Radius of the circle: 15 inches
2. Substitute the given values into the formula:
[tex]\[ \text{Arc Length} = 15 \times \frac{5\pi}{6} \][/tex]
3. Calculate the arc length:
- First, multiply the fraction: [tex]\( \frac{5\pi}{6} \times 15 = \frac{75\pi}{6} \)[/tex]
- Simplify: [tex]\( \frac{75}{6} = 12.5 \)[/tex]
- So, [tex]\( 12.5 \times \pi \approx 39.3 \, \text{inches} \)[/tex] when using [tex]\(\pi \approx 3.14159\)[/tex].
Therefore, the length of the intercepted arc is approximately 39.3 inches, which corresponds to option D.
[tex]\[ \text{Arc Length} = \text{Radius} \times \text{Central Angle (in radians)} \][/tex]
Here are the steps to find the arc length:
1. Identify the given values:
- Central angle: [tex]\(\frac{5\pi}{6}\)[/tex] radians
- Radius of the circle: 15 inches
2. Substitute the given values into the formula:
[tex]\[ \text{Arc Length} = 15 \times \frac{5\pi}{6} \][/tex]
3. Calculate the arc length:
- First, multiply the fraction: [tex]\( \frac{5\pi}{6} \times 15 = \frac{75\pi}{6} \)[/tex]
- Simplify: [tex]\( \frac{75}{6} = 12.5 \)[/tex]
- So, [tex]\( 12.5 \times \pi \approx 39.3 \, \text{inches} \)[/tex] when using [tex]\(\pi \approx 3.14159\)[/tex].
Therefore, the length of the intercepted arc is approximately 39.3 inches, which corresponds to option D.
