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 solve the problem of finding 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, follow these steps:
1. Understand the formula for arc length:
The length of an arc ([tex]\(L\)[/tex]) in a circle can be determined using the formula:
[tex]\[ L = r \theta \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle and [tex]\( \theta \)[/tex] is the central angle in radians.
2. Identify the given values:
- The radius [tex]\( r \)[/tex] is 10 inches.
- The central angle [tex]\( \theta \)[/tex] is [tex]\(\frac{2 \pi}{3}\)[/tex] radians.
3. Substitute the values into the formula:
[tex]\[ L = 10 \times \frac{2\pi}{3} \][/tex]
4. Calculate the arc length:
[tex]\[ L = 10 \times \frac{2 \pi}{3} \][/tex]
[tex]\[ L = \frac{20 \pi}{3} \][/tex]
5. Approximate [tex]\(\pi\)[/tex] to its numerical value:
[tex]\(\pi \approx 3.14159\)[/tex]
6. Perform the multiplication:
[tex]\[ L \approx \frac{20 \times 3.14159}{3} \][/tex]
[tex]\[ L \approx \frac{62.8318}{3} \][/tex]
[tex]\[ L \approx 20.94395 \][/tex]
So, the approximate length of the arc is:
[tex]\[ \boxed{20.94} \, \text{inches} \][/tex]
Therefore, the correct answer from the given choices is:
[tex]\[ 20.94 \, \text{inches} \][/tex]
1. Understand the formula for arc length:
The length of an arc ([tex]\(L\)[/tex]) in a circle can be determined using the formula:
[tex]\[ L = r \theta \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle and [tex]\( \theta \)[/tex] is the central angle in radians.
2. Identify the given values:
- The radius [tex]\( r \)[/tex] is 10 inches.
- The central angle [tex]\( \theta \)[/tex] is [tex]\(\frac{2 \pi}{3}\)[/tex] radians.
3. Substitute the values into the formula:
[tex]\[ L = 10 \times \frac{2\pi}{3} \][/tex]
4. Calculate the arc length:
[tex]\[ L = 10 \times \frac{2 \pi}{3} \][/tex]
[tex]\[ L = \frac{20 \pi}{3} \][/tex]
5. Approximate [tex]\(\pi\)[/tex] to its numerical value:
[tex]\(\pi \approx 3.14159\)[/tex]
6. Perform the multiplication:
[tex]\[ L \approx \frac{20 \times 3.14159}{3} \][/tex]
[tex]\[ L \approx \frac{62.8318}{3} \][/tex]
[tex]\[ L \approx 20.94395 \][/tex]
So, the approximate length of the arc is:
[tex]\[ \boxed{20.94} \, \text{inches} \][/tex]
Therefore, the correct answer from the given choices is:
[tex]\[ 20.94 \, \text{inches} \][/tex]
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