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 length of the arc intersected by a central angle of [tex]\(\frac{2 \pi}{3}\)[/tex] in a circle with a radius of 10 inches, we can use the arc length formula. This formula helps us determine the length of an arc, which is part of the circle's circumference.
The formula for the arc length ([tex]\(L\)[/tex]) is:
[tex]\[ L = r \times \theta \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle,
- [tex]\( \theta \)[/tex] is the central angle in radians.
For this problem:
- The radius [tex]\( r \)[/tex] is 10 inches.
- The central angle [tex]\( \theta \)[/tex] is [tex]\(\frac{2 \pi}{3}\)[/tex] radians.
Plug these values into the formula:
[tex]\[ L = 10 \times \frac{2 \pi}{3} \][/tex]
Now, let's simplify this expression:
1. Multiply 10 by [tex]\(\frac{2 \pi}{3}\)[/tex]:
[tex]\[
L = \left( \frac{10 \times 2 \pi}{3} \right)
\][/tex]
[tex]\[
L = \frac{20 \pi}{3}
\][/tex]
2. To find the approximate numerical value of [tex]\( L \)[/tex], use the approximation [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[
L \approx \frac{20 \times 3.14159}{3}
\][/tex]
[tex]\[
L \approx \frac{62.8318}{3}
\][/tex]
[tex]\[
L \approx 20.94395102393195
\][/tex]
The approximate length of the arc is around 20.94 inches.
Thus, the correct answer is approximately [tex]\( 20.94 \)[/tex] inches, which matches the answer choice:
20.94 inches.
The formula for the arc length ([tex]\(L\)[/tex]) is:
[tex]\[ L = r \times \theta \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle,
- [tex]\( \theta \)[/tex] is the central angle in radians.
For this problem:
- The radius [tex]\( r \)[/tex] is 10 inches.
- The central angle [tex]\( \theta \)[/tex] is [tex]\(\frac{2 \pi}{3}\)[/tex] radians.
Plug these values into the formula:
[tex]\[ L = 10 \times \frac{2 \pi}{3} \][/tex]
Now, let's simplify this expression:
1. Multiply 10 by [tex]\(\frac{2 \pi}{3}\)[/tex]:
[tex]\[
L = \left( \frac{10 \times 2 \pi}{3} \right)
\][/tex]
[tex]\[
L = \frac{20 \pi}{3}
\][/tex]
2. To find the approximate numerical value of [tex]\( L \)[/tex], use the approximation [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[
L \approx \frac{20 \times 3.14159}{3}
\][/tex]
[tex]\[
L \approx \frac{62.8318}{3}
\][/tex]
[tex]\[
L \approx 20.94395102393195
\][/tex]
The approximate length of the arc is around 20.94 inches.
Thus, the correct answer is approximately [tex]\( 20.94 \)[/tex] inches, which matches the answer choice:
20.94 inches.
Thank you for reading the article 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. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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