Thank you for visiting A circle has a radius of 10 inches Find the approximate length of the arc intersected by a central angle of tex frac 2 pi. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
To find the approximate length of the arc intersected by a central angle in a circle, we can use the formula for arc length:
[tex]\[ \text{Arc Length} = \text{Radius} \times \text{Central Angle (in radians)} \][/tex]
Here’s a step-by-step breakdown:
1. Identify the Given Values:
- Radius of the circle, [tex]\( r = 10 \)[/tex] inches.
- Central angle, [tex]\( \theta = \frac{2\pi}{3} \)[/tex] radians.
2. Apply the Arc Length Formula:
Substitute the given values into the formula:
[tex]\[
\text{Arc Length} = 10 \times \frac{2\pi}{3}
\][/tex]
3. Calculate the Arc Length:
- First, calculate [tex]\( \frac{2\pi}{3} \)[/tex]. Using the approximation [tex]\( \pi \approx 3.14159 \)[/tex]:
[tex]\[
\frac{2\pi}{3} \approx \frac{2 \times 3.14159}{3} \approx 2.094395
\][/tex]
- Multiply this by the radius:
[tex]\[
10 \times 2.094395 \approx 20.944
\][/tex]
So, the approximate length of the arc is [tex]\( 20.94 \)[/tex] inches, which matches the choice 20.94 inches from the given options.
[tex]\[ \text{Arc Length} = \text{Radius} \times \text{Central Angle (in radians)} \][/tex]
Here’s a step-by-step breakdown:
1. Identify the Given Values:
- Radius of the circle, [tex]\( r = 10 \)[/tex] inches.
- Central angle, [tex]\( \theta = \frac{2\pi}{3} \)[/tex] radians.
2. Apply the Arc Length Formula:
Substitute the given values into the formula:
[tex]\[
\text{Arc Length} = 10 \times \frac{2\pi}{3}
\][/tex]
3. Calculate the Arc Length:
- First, calculate [tex]\( \frac{2\pi}{3} \)[/tex]. Using the approximation [tex]\( \pi \approx 3.14159 \)[/tex]:
[tex]\[
\frac{2\pi}{3} \approx \frac{2 \times 3.14159}{3} \approx 2.094395
\][/tex]
- Multiply this by the radius:
[tex]\[
10 \times 2.094395 \approx 20.944
\][/tex]
So, the approximate length of the arc is [tex]\( 20.94 \)[/tex] inches, which matches the choice 20.94 inches from the given options.
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Rewritten by : Jeany
