Elementary Algebra
17th Edition
ISBN:9780998625713
Author:Lynn Marecek, MaryAnne Anthony-Smith
Publisher:Lynn Marecek, MaryAnne Anthony-Smith
Chapter8: Rational Expressions And Equations
Section8.9: Use Direct And Inverse Variation
Problem 508E: The area of the face of a Ferris wheel varies directly with the square of its radius. If the area of...
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![**Problem 21 Explanation and Solution**
**Problem Statement:**
Find the area of the shaded region. Round your answer to the nearest tenth.
**Description and Diagram Analysis:**
The provided diagram is a circle with a shaded sector. The central angle of the shaded sector is given as 95°. The radius of the circle is presented as \( 5\sqrt{2} \) meters.
**Solution Steps:**
1. **Calculate the Area of the Entire Circle:**
The formula for the area of a circle is given by:
\[
\text{Area} = \pi r^2
\]
Where \( r \) is the radius of the circle.
Given:
\[
r = 5\sqrt{2} \, \text{m}
\]
Substituting the radius into the formula:
\[
\text{Area} = \pi (5\sqrt{2})^2 = \pi (25 \times 2) = 50\pi \, \text{m}^2
\]
2. **Calculate the Area of the Shaded Sector:**
The area of a sector of a circle is given by:
\[
\text{Sector Area} = \left( \frac{\theta}{360} \right) \times \text{Area of the Circle}
\]
Where \( \theta \) is the central angle of the sector in degrees.
Given:
\[
\theta = 95°
\]
Substituting the values:
\[
\text{Sector Area} = \left( \frac{95}{360} \right) \times 50\pi
\]
Calculate the fraction:
\[
\frac{95}{360} \approx 0.2639
\]
Multiplying with the area of the circle:
\[
\text{Sector Area} \approx 0.2639 \times 50\pi \approx 13.195 \pi
\]
Using \( \pi \approx 3.1416 \):
\[
\text{Sector Area} \approx 13.195 \times 3.1416 \approx 41.5 \, \text{m}^2
\]
**Final Answer:**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c8a549a-ce13-4c73-8e46-ff62e8d935c2%2F39e8f167-fbbc-4462-98c7-65f7cf8ed59c%2Ffw7zw5n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 21 Explanation and Solution**
**Problem Statement:**
Find the area of the shaded region. Round your answer to the nearest tenth.
**Description and Diagram Analysis:**
The provided diagram is a circle with a shaded sector. The central angle of the shaded sector is given as 95°. The radius of the circle is presented as \( 5\sqrt{2} \) meters.
**Solution Steps:**
1. **Calculate the Area of the Entire Circle:**
The formula for the area of a circle is given by:
\[
\text{Area} = \pi r^2
\]
Where \( r \) is the radius of the circle.
Given:
\[
r = 5\sqrt{2} \, \text{m}
\]
Substituting the radius into the formula:
\[
\text{Area} = \pi (5\sqrt{2})^2 = \pi (25 \times 2) = 50\pi \, \text{m}^2
\]
2. **Calculate the Area of the Shaded Sector:**
The area of a sector of a circle is given by:
\[
\text{Sector Area} = \left( \frac{\theta}{360} \right) \times \text{Area of the Circle}
\]
Where \( \theta \) is the central angle of the sector in degrees.
Given:
\[
\theta = 95°
\]
Substituting the values:
\[
\text{Sector Area} = \left( \frac{95}{360} \right) \times 50\pi
\]
Calculate the fraction:
\[
\frac{95}{360} \approx 0.2639
\]
Multiplying with the area of the circle:
\[
\text{Sector Area} \approx 0.2639 \times 50\pi \approx 13.195 \pi
\]
Using \( \pi \approx 3.1416 \):
\[
\text{Sector Area} \approx 13.195 \times 3.1416 \approx 41.5 \, \text{m}^2
\]
**Final Answer:**
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