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Answer :
Sure! Let's find the approximate length of the arc intersected by a central angle of [tex]\(\frac{2 \pi}{3}\)[/tex] radians for a circle with a radius of 10 inches.
Here's a step-by-step solution to the problem:
1. Understand the problem:
- We have a circle with a radius of 10 inches.
- The central angle is [tex]\(\frac{2 \pi}{3}\)[/tex] radians.
- We need to find the length of the arc that is intercepted by the given central angle.
2. Formula for arc length:
The length of an arc ([tex]\(L\)[/tex]) of a circle can be computed using the formula:
[tex]\[
L = r \theta
\][/tex]
where:
- [tex]\(r\)[/tex] is the radius of the circle,
- [tex]\(\theta\)[/tex] is the central angle in radians.
3. Substitute the given values:
Here, [tex]\(r = 10\)[/tex] inches and [tex]\(\theta = \frac{2 \pi}{3}\)[/tex] radians.
4. Compute the arc length:
[tex]\[
L = 10 \times \frac{2 \pi}{3}
\][/tex]
5. Simplify the expression:
[tex]\[
L = \frac{20 \pi}{3}
\][/tex]
6. Approximate the value using [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[
L \approx \frac{20 \times 3.14159}{3} \approx 20.94395 \text{ inches}
\][/tex]
So, the approximate length of the arc is 20.94 inches.
From the given options, the correct answer is:
- 20.94 inches
Therefore, the length of the arc is approximately 20.94 inches.
Here's a step-by-step solution to the problem:
1. Understand the problem:
- We have a circle with a radius of 10 inches.
- The central angle is [tex]\(\frac{2 \pi}{3}\)[/tex] radians.
- We need to find the length of the arc that is intercepted by the given central angle.
2. Formula for arc length:
The length of an arc ([tex]\(L\)[/tex]) of a circle can be computed using the formula:
[tex]\[
L = r \theta
\][/tex]
where:
- [tex]\(r\)[/tex] is the radius of the circle,
- [tex]\(\theta\)[/tex] is the central angle in radians.
3. Substitute the given values:
Here, [tex]\(r = 10\)[/tex] inches and [tex]\(\theta = \frac{2 \pi}{3}\)[/tex] radians.
4. Compute the arc length:
[tex]\[
L = 10 \times \frac{2 \pi}{3}
\][/tex]
5. Simplify the expression:
[tex]\[
L = \frac{20 \pi}{3}
\][/tex]
6. Approximate the value using [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[
L \approx \frac{20 \times 3.14159}{3} \approx 20.94395 \text{ inches}
\][/tex]
So, the approximate length of the arc is 20.94 inches.
From the given options, the correct answer is:
- 20.94 inches
Therefore, the length of the arc is approximately 20.94 inches.
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