An area of a circular sector is 50m? with radius of 5 m. Find the length of the same sector.
An area of a circular sector is 50m? with radius of 5 m. Find the length of the same sector.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
![**Problem Statement:**
An area of a circular sector is \(50 \, \text{m}^2\) with a radius of \(5 \, \text{m}\). Find the length of the same sector.
**Solution Explanation:**
To find the length of the sector, first, we need to determine the angle of the sector in radians. The area \(A\) of a circular sector is given by the formula:
\[ A = \frac{1}{2} \times r^2 \times \theta \]
Where:
- \( A \) is the area of the sector,
- \( r \) is the radius,
- \( \theta \) is the angle in radians.
Given:
\[ A = 50 \, \text{m}^2 \]
\[ r = 5 \, \text{m} \]
Substitute the known values into the formula and solve for \(\theta\):
\[ 50 = \frac{1}{2} \times 5^2 \times \theta \]
\[ 50 = \frac{1}{2} \times 25 \times \theta \]
\[ 50 = 12.5 \times \theta \]
\[ \theta = \frac{50}{12.5} \]
\[ \theta = 4 \, \text{radians} \]
Now, to find the arc length \(L\), use the formula:
\[ L = r \times \theta \]
Substitute the known values:
\[ L = 5 \times 4 \]
\[ L = 20 \, \text{m} \]
Therefore, the length of the sector is \(20 \, \text{m}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b727f25-8808-4b74-9fb9-f2e74344edb6%2F24c02928-0bab-42b5-ab33-f73a97432e93%2Fdzvl2kf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
An area of a circular sector is \(50 \, \text{m}^2\) with a radius of \(5 \, \text{m}\). Find the length of the same sector.
**Solution Explanation:**
To find the length of the sector, first, we need to determine the angle of the sector in radians. The area \(A\) of a circular sector is given by the formula:
\[ A = \frac{1}{2} \times r^2 \times \theta \]
Where:
- \( A \) is the area of the sector,
- \( r \) is the radius,
- \( \theta \) is the angle in radians.
Given:
\[ A = 50 \, \text{m}^2 \]
\[ r = 5 \, \text{m} \]
Substitute the known values into the formula and solve for \(\theta\):
\[ 50 = \frac{1}{2} \times 5^2 \times \theta \]
\[ 50 = \frac{1}{2} \times 25 \times \theta \]
\[ 50 = 12.5 \times \theta \]
\[ \theta = \frac{50}{12.5} \]
\[ \theta = 4 \, \text{radians} \]
Now, to find the arc length \(L\), use the formula:
\[ L = r \times \theta \]
Substitute the known values:
\[ L = 5 \times 4 \]
\[ L = 20 \, \text{m} \]
Therefore, the length of the sector is \(20 \, \text{m}\).
Expert Solution
Step 1
The given data is:
The area of the circular sector is
To find the length of the same sector.
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