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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Question
![**Problem 5: Finding the Area of a Sector**
**Problem Statement:**
In circle \( J \) with \( m \angle HJK = 54^\circ \) and \( HJ = 14 \), find the area of sector \( HJK \). Round to the nearest hundredth.
**Detailed Diagram Explanation:**
In the provided diagram, there is a circle with center \( J \). Points \( H \) and \( K \) lie on the circumference of the circle. The angle subtended by the arc \( HK \) at the center \( J \) is given as \( 54^\circ \). Additionally, the radius of the circle, represented as \( HJ \), is labeled as 14 units.
**Steps to Solve:**
1. **Identify the Given Elements:**
- Central angle \( \angle HJK = 54^\circ \)
- Radius \( HJ = 14 \)
2. **Calculate the Area of the Full Circle:**
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
Here, \( r \) is the radius. Substituting the given radius:
\[
A = \pi \times 14^2
\]
\[
A = 196\pi
\]
3. **Find the Fraction of the Circle Represented by the Sector:**
The fraction of the circle's area that the sector occupies is determined by the ratio of the central angle to the full angle of a circle (360 degrees):
\[
\text{Fraction} = \frac{54^\circ}{360^\circ}
\]
Simplifying, we get:
\[
\text{Fraction} = \frac{3}{20}
\]
4. **Calculate the Area of the Sector:**
The area of sector \( HJK \) is found by multiplying the fraction by the total area of the circle:
\[
\text{Area of Sector} = \frac{3}{20} \times 196\pi
\]
\[
\text{Area of Sector} = \frac{3 \times 196\pi}{20}
\]
\[
\text{Area of Sector} = \frac{588\pi}{20](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F352b6feb-25e0-422d-9e10-43745f0b2a3e%2F2a9382b3-ea67-4ae6-886f-e93b0ebf9bf8%2Fc9t5hw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5: Finding the Area of a Sector**
**Problem Statement:**
In circle \( J \) with \( m \angle HJK = 54^\circ \) and \( HJ = 14 \), find the area of sector \( HJK \). Round to the nearest hundredth.
**Detailed Diagram Explanation:**
In the provided diagram, there is a circle with center \( J \). Points \( H \) and \( K \) lie on the circumference of the circle. The angle subtended by the arc \( HK \) at the center \( J \) is given as \( 54^\circ \). Additionally, the radius of the circle, represented as \( HJ \), is labeled as 14 units.
**Steps to Solve:**
1. **Identify the Given Elements:**
- Central angle \( \angle HJK = 54^\circ \)
- Radius \( HJ = 14 \)
2. **Calculate the Area of the Full Circle:**
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
Here, \( r \) is the radius. Substituting the given radius:
\[
A = \pi \times 14^2
\]
\[
A = 196\pi
\]
3. **Find the Fraction of the Circle Represented by the Sector:**
The fraction of the circle's area that the sector occupies is determined by the ratio of the central angle to the full angle of a circle (360 degrees):
\[
\text{Fraction} = \frac{54^\circ}{360^\circ}
\]
Simplifying, we get:
\[
\text{Fraction} = \frac{3}{20}
\]
4. **Calculate the Area of the Sector:**
The area of sector \( HJK \) is found by multiplying the fraction by the total area of the circle:
\[
\text{Area of Sector} = \frac{3}{20} \times 196\pi
\]
\[
\text{Area of Sector} = \frac{3 \times 196\pi}{20}
\]
\[
\text{Area of Sector} = \frac{588\pi}{20
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