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 of [tex]\(\frac{2\pi}{3}\)[/tex] in a circle with a radius of 10 inches, we can follow these steps:
1. Understand the Geometry: The problem involves a circle and a central angle. The arc length is part of the circumference of the circle. The entire circumference of the circle is given by the formula [tex]\(C = 2\pi \times \text{radius}\)[/tex].
2. Calculate the Total Circumference: For this circle, with a radius of 10 inches, the total circumference is:
[tex]\[
C = 2\pi \times 10 = 20\pi \text{ inches}
\][/tex]
3. Utilize the Central Angle: The arc length is directly proportional to the central angle. The entire circle encompasses an angle of [tex]\(2\pi\)[/tex] radians. Here, we are interested in a central angle of [tex]\(\frac{2\pi}{3}\)[/tex].
4. Determine the Proportion of the Angle to the Circle: The portion of the circle's circumference that corresponds to the arc is determined by the ratio of the central angle to the full angle in a circle:
[tex]\[
\text{Proportion} = \frac{\frac{2\pi}{3}}{2\pi} = \frac{2}{3}
\][/tex]
5. Calculate the Arc Length: Finally, apply this proportion to the total circumference to find the arc length:
[tex]\[
\text{Arc Length} = C \times \text{Proportion} = 20\pi \times \frac{2}{3} = \frac{40\pi}{3} \text{ inches}
\][/tex]
6. Approximate the Arc Length: To get an approximate numerical value of the arc length, use the approximation [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[
\text{Arc Length} \approx \frac{40 \times 3.14159}{3} \approx 20.94 \text{ inches}
\][/tex]
Therefore, the approximate length of the arc is 20.94 inches. This matches with the first choice provided in the answer options.
1. Understand the Geometry: The problem involves a circle and a central angle. The arc length is part of the circumference of the circle. The entire circumference of the circle is given by the formula [tex]\(C = 2\pi \times \text{radius}\)[/tex].
2. Calculate the Total Circumference: For this circle, with a radius of 10 inches, the total circumference is:
[tex]\[
C = 2\pi \times 10 = 20\pi \text{ inches}
\][/tex]
3. Utilize the Central Angle: The arc length is directly proportional to the central angle. The entire circle encompasses an angle of [tex]\(2\pi\)[/tex] radians. Here, we are interested in a central angle of [tex]\(\frac{2\pi}{3}\)[/tex].
4. Determine the Proportion of the Angle to the Circle: The portion of the circle's circumference that corresponds to the arc is determined by the ratio of the central angle to the full angle in a circle:
[tex]\[
\text{Proportion} = \frac{\frac{2\pi}{3}}{2\pi} = \frac{2}{3}
\][/tex]
5. Calculate the Arc Length: Finally, apply this proportion to the total circumference to find the arc length:
[tex]\[
\text{Arc Length} = C \times \text{Proportion} = 20\pi \times \frac{2}{3} = \frac{40\pi}{3} \text{ inches}
\][/tex]
6. Approximate the Arc Length: To get an approximate numerical value of the arc length, use the approximation [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[
\text{Arc Length} \approx \frac{40 \times 3.14159}{3} \approx 20.94 \text{ inches}
\][/tex]
Therefore, the approximate length of the arc is 20.94 inches. This matches with the first choice provided in the answer options.
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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