Answer :
To find the length of the intercepted arc when you have a central angle and a radius, you can use the formula for arc length:
[tex]\[ \text{Arc Length} = \theta \times r \][/tex]
where:
- [tex]\(\theta\)[/tex] is the central angle in radians,
- [tex]\(r\)[/tex] is the radius of the circle.
In this problem:
- The central angle [tex]\(\theta\)[/tex] is [tex]\(\frac{5\pi}{6}\)[/tex].
- The radius [tex]\(r\)[/tex] is 15 inches.
Substitute these values into the formula:
[tex]\[ \text{Arc Length} = \frac{5\pi}{6} \times 15 \][/tex]
Now, calculate:
1. Multiply the fractions:
- [tex]\(\frac{5}{6} \times 15 = \frac{75}{6} = 12.5\)[/tex]
2. Multiply by [tex]\(\pi\)[/tex]:
- [tex]\(12.5 \times \pi\)[/tex]
3. Using the approximation [tex]\(\pi \approx 3.14159\)[/tex]:
- [tex]\(12.5 \times 3.14159 \approx 39.27\)[/tex]
The length of the intercepted arc is approximately 39.27 inches. Comparing this value to the given options, the closest answer is:
d. 39.3 inches
So, the best choice is D.
[tex]\[ \text{Arc Length} = \theta \times r \][/tex]
where:
- [tex]\(\theta\)[/tex] is the central angle in radians,
- [tex]\(r\)[/tex] is the radius of the circle.
In this problem:
- The central angle [tex]\(\theta\)[/tex] is [tex]\(\frac{5\pi}{6}\)[/tex].
- The radius [tex]\(r\)[/tex] is 15 inches.
Substitute these values into the formula:
[tex]\[ \text{Arc Length} = \frac{5\pi}{6} \times 15 \][/tex]
Now, calculate:
1. Multiply the fractions:
- [tex]\(\frac{5}{6} \times 15 = \frac{75}{6} = 12.5\)[/tex]
2. Multiply by [tex]\(\pi\)[/tex]:
- [tex]\(12.5 \times \pi\)[/tex]
3. Using the approximation [tex]\(\pi \approx 3.14159\)[/tex]:
- [tex]\(12.5 \times 3.14159 \approx 39.27\)[/tex]
The length of the intercepted arc is approximately 39.27 inches. Comparing this value to the given options, the closest answer is:
d. 39.3 inches
So, the best choice is D.
