Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
![**Mathematics Problem: Calculating the Area of a Shaded Region**
**Problem 12: What is the area of the shaded area below? Answer in terms of π.**
**Given Information:**
- Diagram Description: There are two concentric circles (one circle within another).
- The radius of the larger circle: 8 cm
- The radius of the smaller circle: 3 cm
**Step-by-Step Solution:**
To find the area of the shaded region, we need to subtract the area of the smaller circle from the area of the larger circle.
1. **Calculate the area of the larger circle:**
The area \(A\) of a circle is given by the formula:
\[
A = \pi r^2
\]
where \(r\) is the radius.
For the larger circle:
\[
\text{Radius} = 8 \text{ cm}
\]
\[
A_{\text{large}} = \pi (8)^2 = 64\pi \text{ cm}^2
\]
2. **Calculate the area of the smaller circle:**
For the smaller circle:
\[
\text{Radius} = 3 \text{ cm}
\]
\[
A_{\text{small}} = \pi (3)^2 = 9\pi \text{ cm}^2
\]
3. **Determine the area of the shaded region:**
\[
A_{\text{shaded}} = A_{\text{large}} - A_{\text{small}}
\]
\[
A_{\text{shaded}} = 64\pi - 9\pi = 55\pi \text{ cm}^2
\]
Thus, the area of the shaded region is \(55\pi\) square centimeters.
**Final Answer:**
Area = \(55\pi \text{ cm}^2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F575cb657-4d5f-4fcc-8996-f1fa43fab624%2Fbf43c1c5-b41b-433a-8be0-ddf9c76509c4%2Fungnuc6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics Problem: Calculating the Area of a Shaded Region**
**Problem 12: What is the area of the shaded area below? Answer in terms of π.**
**Given Information:**
- Diagram Description: There are two concentric circles (one circle within another).
- The radius of the larger circle: 8 cm
- The radius of the smaller circle: 3 cm
**Step-by-Step Solution:**
To find the area of the shaded region, we need to subtract the area of the smaller circle from the area of the larger circle.
1. **Calculate the area of the larger circle:**
The area \(A\) of a circle is given by the formula:
\[
A = \pi r^2
\]
where \(r\) is the radius.
For the larger circle:
\[
\text{Radius} = 8 \text{ cm}
\]
\[
A_{\text{large}} = \pi (8)^2 = 64\pi \text{ cm}^2
\]
2. **Calculate the area of the smaller circle:**
For the smaller circle:
\[
\text{Radius} = 3 \text{ cm}
\]
\[
A_{\text{small}} = \pi (3)^2 = 9\pi \text{ cm}^2
\]
3. **Determine the area of the shaded region:**
\[
A_{\text{shaded}} = A_{\text{large}} - A_{\text{small}}
\]
\[
A_{\text{shaded}} = 64\pi - 9\pi = 55\pi \text{ cm}^2
\]
Thus, the area of the shaded region is \(55\pi\) square centimeters.
**Final Answer:**
Area = \(55\pi \text{ cm}^2\).
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