Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Arc Length Calculation
**Problem:**
23. What is the length of an arc having a radius of curvature of 12 and an angle measure of 226 degrees?
**Solution Explanation:**
To find the length of an arc (L) with a given radius (r) and central angle (θ), you can use the formula:
\[ L = r \times \theta \]
However, the angle θ must be in radians. To convert degrees to radians, use the conversion factor:
\[ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right) \]
Applying this conversion to the given angle of 226 degrees:
\[ \theta = 226 \times \left( \frac{\pi}{180} \right) \]
\[ \theta = 226 \times \left( \frac{3.14159}{180} \right) \]
\[ \theta \approx 3.945 \, \text{radians} \]
Now, using the radius \( r = 12 \):
\[ L = 12 \times 3.945 \]
\[ L \approx 47.34 \]
Therefore, the length of the arc is approximately 47.34 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba874e82-f331-4b1e-9400-f6613fffe075%2F577c28bb-976a-4a2d-aa8b-278f99d1b6a5%2F4lnj7x7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Arc Length Calculation
**Problem:**
23. What is the length of an arc having a radius of curvature of 12 and an angle measure of 226 degrees?
**Solution Explanation:**
To find the length of an arc (L) with a given radius (r) and central angle (θ), you can use the formula:
\[ L = r \times \theta \]
However, the angle θ must be in radians. To convert degrees to radians, use the conversion factor:
\[ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right) \]
Applying this conversion to the given angle of 226 degrees:
\[ \theta = 226 \times \left( \frac{\pi}{180} \right) \]
\[ \theta = 226 \times \left( \frac{3.14159}{180} \right) \]
\[ \theta \approx 3.945 \, \text{radians} \]
Now, using the radius \( r = 12 \):
\[ L = 12 \times 3.945 \]
\[ L \approx 47.34 \]
Therefore, the length of the arc is approximately 47.34 units.
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