the area of the shaded part If x-5. Round to the nearest 60
A First Course in Probability (10th Edition)
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Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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![### Problem 7: Shaded Area Calculation
**Question:**
What is the area of the shaded part of circle \( \Theta K \). If \( x = 5 \). Round to the nearest tenth.
**Diagram Explanation:**
The diagram shows a circle with center \( K \) and radius \( x \) (where \( x = 5 \)). A sector \( M \angle K L \) with a central angle of \( 60^\circ \) is shaded.
**Solution Steps:**
1. **Find the Area of the Circle:**
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
Where \( r \) is the radius. Given \( x = 5 \):
\[
A = \pi (5)^2 = 25\pi
\]
2. **Find the Area of the Sector:**
The area of a sector is a fraction of the circle's area. The fraction is determined by the central angle \( \theta \):
\[
\text{Sector Area} = \left( \frac{\theta}{360} \right) \times \text{Area of the Circle}
\]
Given the central angle is \( 60^\circ \):
\[
\text{Sector Area} = \left( \frac{60}{360} \right) \times 25\pi = \left( \frac{1}{6} \right) \times 25\pi = \frac{25\pi}{6}
\]
Calculate the numerical value:
\[
\text{Sector Area} = \frac{25\pi}{6} \approx \frac{25 \times 3.1416}{6} \approx 13.1
\]
Thus, the area of the shaded part of circle \( \Theta K \) is approximately \( 13.1 \) square units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6d2261f-dbba-4874-8d7e-5ac5e2791464%2F4b0ee0fa-65fd-4fb6-8cb9-3010c8679d87%2F336hz2h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 7: Shaded Area Calculation
**Question:**
What is the area of the shaded part of circle \( \Theta K \). If \( x = 5 \). Round to the nearest tenth.
**Diagram Explanation:**
The diagram shows a circle with center \( K \) and radius \( x \) (where \( x = 5 \)). A sector \( M \angle K L \) with a central angle of \( 60^\circ \) is shaded.
**Solution Steps:**
1. **Find the Area of the Circle:**
The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
Where \( r \) is the radius. Given \( x = 5 \):
\[
A = \pi (5)^2 = 25\pi
\]
2. **Find the Area of the Sector:**
The area of a sector is a fraction of the circle's area. The fraction is determined by the central angle \( \theta \):
\[
\text{Sector Area} = \left( \frac{\theta}{360} \right) \times \text{Area of the Circle}
\]
Given the central angle is \( 60^\circ \):
\[
\text{Sector Area} = \left( \frac{60}{360} \right) \times 25\pi = \left( \frac{1}{6} \right) \times 25\pi = \frac{25\pi}{6}
\]
Calculate the numerical value:
\[
\text{Sector Area} = \frac{25\pi}{6} \approx \frac{25 \times 3.1416}{6} \approx 13.1
\]
Thus, the area of the shaded part of circle \( \Theta K \) is approximately \( 13.1 \) square units.
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