Find the length s and area A. The length of the arc is yards. (Round to three decimal places as needed.) The area A of the sector is square yards. (Round to three decimal places as needed.)
Find the length s and area A. The length of the arc is yards. (Round to three decimal places as needed.) The area A of the sector is square yards. (Round to three decimal places as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Exploration of Arc Length and Sector Area Calculation**
In our exploration today, we will focus on finding the length \(s\) of an arc and the area \(A\) of a sector. This will involve working with circular arcs and understanding how to measure and calculate these attributes accurately.
**Task:**
- **Calculate the length \(s\) of the arc.**
Ensure that your calculation is precise, rounding to three decimal places as necessary.
- **Determine the area \(A\) of the sector.**
As with the arc length, round your result to three decimal places.
**Guidance:**
- **Arc Length Calculation:**
To find the arc length \(s\), use the formula:
\[
s = r \theta
\]
where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians.
- **Sector Area Calculation:**
The area \(A\) of the sector can be determined using:
\[
A = \frac{1}{2} r^2 \theta
\]
This formula also requires the radius \(r\) and the central angle \(\theta\) in radians.
By mastering these calculations, you will be equipped to solve real-world problems involving circular measurements. This understanding is essential for various applications in fields such as engineering, architecture, and mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F047d3997-5560-45e5-ac13-3cddd4f35842%2Ff644452d-dbfc-46c9-b30c-216bcf7a6742%2Fzhp9jme_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exploration of Arc Length and Sector Area Calculation**
In our exploration today, we will focus on finding the length \(s\) of an arc and the area \(A\) of a sector. This will involve working with circular arcs and understanding how to measure and calculate these attributes accurately.
**Task:**
- **Calculate the length \(s\) of the arc.**
Ensure that your calculation is precise, rounding to three decimal places as necessary.
- **Determine the area \(A\) of the sector.**
As with the arc length, round your result to three decimal places.
**Guidance:**
- **Arc Length Calculation:**
To find the arc length \(s\), use the formula:
\[
s = r \theta
\]
where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians.
- **Sector Area Calculation:**
The area \(A\) of the sector can be determined using:
\[
A = \frac{1}{2} r^2 \theta
\]
This formula also requires the radius \(r\) and the central angle \(\theta\) in radians.
By mastering these calculations, you will be equipped to solve real-world problems involving circular measurements. This understanding is essential for various applications in fields such as engineering, architecture, and mathematics.
Transcribed Image Text:The image shows a circle with a highlighted sector. This sector is labeled with an angle of 20° at the center, denoted by point A. The radius of the circle is labeled as 14 yards. The segment labeled "s" represents the arc of the sector. The sector is shaded and bordered with a thicker line indicating its distinction from the rest of the circle.
Key components of the diagram:
- **Circle**: The entire figure is a circle with a marked sector.
- **Sector**: The portion of the circle bounded by two radii and the arc. It is highlighted in a different shade.
- **Angle**: The central angle of the sector, 20°, indicating the extent of the arc.
- **Radius**: The distance from the center of the circle to the edge, given as 14 yards.
- **Arc "s"**: The curved part of the sector which is opposite the central angle.
This diagram can help in understanding concepts related to circles such as circumference, arc length, and area of a sector.
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