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
![**Question 15**
**Problem Statement:**
Find the area of the shaded area. Leave your answers in terms of π.
**Explanation:**
The diagram shows two concentric circles. The radius of the smaller circle is 5 units, and the radius of the larger circle is 6 units.
To find the area of the shaded region, we need to:
1. Calculate the area of the larger circle.
2. Calculate the area of the smaller circle.
3. Subtract the area of the smaller circle from the area of the larger circle.
**Formulas:**
The area of a circle is given by:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.
**Calculations:**
1. Calculate the area of the larger circle (radius \( r = 6 \)):
\[ A_{\text{large}} = \pi \times 6^2 = 36\pi \text{ square units} \]
2. Calculate the area of the smaller circle (radius \( r = 5 \)):
\[ A_{\text{small}} = \pi \times 5^2 = 25\pi \text{ square units} \]
3. Subtract the area of the smaller circle from the area of the larger circle to get the shaded area:
\[ A_{\text{shaded}} = A_{\text{large}} - A_{\text{small}} = 36\pi - 25\pi = 11\pi \text{ square units} \]
Therefore, the area of the shaded region is \( 11\pi \) square units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7c29d2f-5cf3-43f5-aa90-abc5a822140a%2F32990020-c502-45d3-9732-b25675bc9490%2F8i7jtcl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 15**
**Problem Statement:**
Find the area of the shaded area. Leave your answers in terms of π.
**Explanation:**
The diagram shows two concentric circles. The radius of the smaller circle is 5 units, and the radius of the larger circle is 6 units.
To find the area of the shaded region, we need to:
1. Calculate the area of the larger circle.
2. Calculate the area of the smaller circle.
3. Subtract the area of the smaller circle from the area of the larger circle.
**Formulas:**
The area of a circle is given by:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.
**Calculations:**
1. Calculate the area of the larger circle (radius \( r = 6 \)):
\[ A_{\text{large}} = \pi \times 6^2 = 36\pi \text{ square units} \]
2. Calculate the area of the smaller circle (radius \( r = 5 \)):
\[ A_{\text{small}} = \pi \times 5^2 = 25\pi \text{ square units} \]
3. Subtract the area of the smaller circle from the area of the larger circle to get the shaded area:
\[ A_{\text{shaded}} = A_{\text{large}} - A_{\text{small}} = 36\pi - 25\pi = 11\pi \text{ square units} \]
Therefore, the area of the shaded region is \( 11\pi \) square units.
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