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Answer :
To find the length of the arc intersected by a central angle in a circle, we use the formula:
[tex]\[ \text{Arc Length} = \text{Radius} \times \text{Central Angle in Radians} \][/tex]
Let's break it down step-by-step:
1. Identify the radius of the circle:
The radius is given as 10 inches.
2. Identify the central angle in radians:
The central angle is given as [tex]\(\frac{2\pi}{3}\)[/tex] radians.
3. Calculate the arc length:
Substitute the known values into the formula:
[tex]\[
\text{Arc Length} = 10 \times \frac{2\pi}{3}
\][/tex]
4. Solve for the arc length:
First, calculate [tex]\(\frac{2\pi}{3}\)[/tex]:
- [tex]\(\pi \approx 3.14159\)[/tex], so [tex]\(2\pi \approx 6.28318\)[/tex]
Then, find [tex]\( \frac{2\pi}{3} \)[/tex]:
- [tex]\(\frac{6.28318}{3} \approx 2.09439\)[/tex]
Now, multiply by the radius:
- [tex]\(10 \times 2.09439 \approx 20.94\)[/tex]
So, the approximate length of the arc is 20.94 inches. This corresponds to the option 20.94 inches in the provided choices.
[tex]\[ \text{Arc Length} = \text{Radius} \times \text{Central Angle in Radians} \][/tex]
Let's break it down step-by-step:
1. Identify the radius of the circle:
The radius is given as 10 inches.
2. Identify the central angle in radians:
The central angle is given as [tex]\(\frac{2\pi}{3}\)[/tex] radians.
3. Calculate the arc length:
Substitute the known values into the formula:
[tex]\[
\text{Arc Length} = 10 \times \frac{2\pi}{3}
\][/tex]
4. Solve for the arc length:
First, calculate [tex]\(\frac{2\pi}{3}\)[/tex]:
- [tex]\(\pi \approx 3.14159\)[/tex], so [tex]\(2\pi \approx 6.28318\)[/tex]
Then, find [tex]\( \frac{2\pi}{3} \)[/tex]:
- [tex]\(\frac{6.28318}{3} \approx 2.09439\)[/tex]
Now, multiply by the radius:
- [tex]\(10 \times 2.09439 \approx 20.94\)[/tex]
So, the approximate length of the arc is 20.94 inches. This corresponds to the option 20.94 inches in the provided choices.
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