Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter2: Right Triangle Trigonometry
Section2.4: Applications
Problem 48PS: Albert lives in New Orleans. At noon on a summer day, the angle of elevation of the sun is 84. The...
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![### Find the Area of the Shaded Sector
This problem involves calculating the area of a shaded sector within a circle.
#### Diagram Explanation:
- The diagram is of a circle centered at point O.
- The radius of the circle is labeled as 10 units.
- An angle of 18 degrees is formed at the center, corresponding to the shaded sector.
#### Steps to Calculate the Area of the Shaded Sector:
1. **Understand the formula**:
The area \( A \) of a sector of a circle can be determined using the following formula:
\[
A = \frac{\theta}{360} \times \pi r^2
\]
where:
- \(\theta\) is the central angle in degrees.
- \(r\) is the radius of the circle.
2. **Substitute the values**:
Here, \(\theta = 18^\circ\) and \(r = 10\).
So, we substitute these values into the formula:
\[
A = \frac{18}{360} \times \pi \times 10^2
\]
3. **Simplify the fractions**:
\[
A = \frac{1}{20} \times \pi \times 100
\]
4. **Calculate the area**:
\[
A = 5\pi
\]
Hence, the area of the shaded sector is \( 5\pi \) square units.
#### Final Answer
The area of the shaded sector is \( 5\pi \) square units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F234dd4ba-addc-435e-a54f-15f12e8dfa5c%2Ff63f1e60-1fe3-4c63-b371-65da49d6dfb8%2Fndyqq6j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Find the Area of the Shaded Sector
This problem involves calculating the area of a shaded sector within a circle.
#### Diagram Explanation:
- The diagram is of a circle centered at point O.
- The radius of the circle is labeled as 10 units.
- An angle of 18 degrees is formed at the center, corresponding to the shaded sector.
#### Steps to Calculate the Area of the Shaded Sector:
1. **Understand the formula**:
The area \( A \) of a sector of a circle can be determined using the following formula:
\[
A = \frac{\theta}{360} \times \pi r^2
\]
where:
- \(\theta\) is the central angle in degrees.
- \(r\) is the radius of the circle.
2. **Substitute the values**:
Here, \(\theta = 18^\circ\) and \(r = 10\).
So, we substitute these values into the formula:
\[
A = \frac{18}{360} \times \pi \times 10^2
\]
3. **Simplify the fractions**:
\[
A = \frac{1}{20} \times \pi \times 100
\]
4. **Calculate the area**:
\[
A = 5\pi
\]
Hence, the area of the shaded sector is \( 5\pi \) square units.
#### Final Answer
The area of the shaded sector is \( 5\pi \) square units.
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