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Answer :
To find the length of the intercepted arc with a central angle of [tex]\(\frac{5 \pi}{6}\)[/tex] radians in a circle with a radius of 15 inches, we can use the formula for the length of an arc:
[tex]\[ \text{Arc Length} = r \times \theta \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle,
- [tex]\( \theta \)[/tex] is the central angle in radians.
Let's go through the steps:
1. Identify the given values:
- Radius ([tex]\( r \)[/tex]) = 15 inches
- Central angle ([tex]\( \theta \)[/tex]) = [tex]\(\frac{5 \pi}{6}\)[/tex] radians
2. Apply the formula:
[tex]\[
\text{Arc Length} = 15 \times \frac{5 \pi}{6}
\][/tex]
3. Calculate:
[tex]\[
\text{Arc Length} = 15 \times \frac{5 \pi}{6} = 15 \times \frac{5 \times 3.14159}{6}
\][/tex]
[tex]\[
\text{Arc Length} \approx 15 \times 2.61799
\][/tex]
[tex]\[
\text{Arc Length} \approx 39.27 \text{ inches}
\][/tex]
Therefore, the length of the intercepted arc is approximately 39.3 inches. The correct answer from the provided choices is:
d. 39.3 inches
[tex]\[ \text{Arc Length} = r \times \theta \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle,
- [tex]\( \theta \)[/tex] is the central angle in radians.
Let's go through the steps:
1. Identify the given values:
- Radius ([tex]\( r \)[/tex]) = 15 inches
- Central angle ([tex]\( \theta \)[/tex]) = [tex]\(\frac{5 \pi}{6}\)[/tex] radians
2. Apply the formula:
[tex]\[
\text{Arc Length} = 15 \times \frac{5 \pi}{6}
\][/tex]
3. Calculate:
[tex]\[
\text{Arc Length} = 15 \times \frac{5 \pi}{6} = 15 \times \frac{5 \times 3.14159}{6}
\][/tex]
[tex]\[
\text{Arc Length} \approx 15 \times 2.61799
\][/tex]
[tex]\[
\text{Arc Length} \approx 39.27 \text{ inches}
\][/tex]
Therefore, the length of the intercepted arc is approximately 39.3 inches. The correct answer from the provided choices is:
d. 39.3 inches
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