A circular flower bed is 16 m in diameter and has a circular sidewalk around it 3 m wide. Find the area of the sidewalk in square meters The area of the walk isO m?. (Type an exact answer, using t as needed.)
A circular flower bed is 16 m in diameter and has a circular sidewalk around it 3 m wide. Find the area of the sidewalk in square meters The area of the walk isO m?. (Type an exact answer, using t as needed.)
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
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Question
![### Problem Statement:
A circular flower bed is 16 meters in diameter and has a circular sidewalk around it that is 3 meters wide. Find the area of the sidewalk in square meters.
#### Solution:
To find the area of the sidewalk, we first need to calculate the area of the larger circle (including the sidewalk) and the area of the flower bed (the smaller circle).
1. **Calculate the radius of the flower bed:**
The diameter of the flower bed is given as 16 meters. Therefore, the radius (r) is half of the diameter.
\[
r_{\text{flower bed}} = \frac{16}{2} = 8 \text{ meters}
\]
2. **Calculate the radius of the larger circle (flower bed + sidewalk):**
The sidewalk is 3 meters wide, so we add this width to the radius of the flower bed.
\[
r_{\text{total}} = r_{\text{flower bed}} + 3 = 8 + 3 = 11 \text{ meters}
\]
3. **Calculate the area of the larger circle:**
Using the formula for the area of a circle, \(A = \pi r^2\),
\[
A_{\text{larger}} = \pi \times 11^2 = \pi \times 121 \text{ square meters}
\]
4. **Calculate the area of the flower bed:**
Again, using the formula for the area of a circle,
\[
A_{\text{flower bed}} = \pi \times 8^2 = \pi \times 64 \text{ square meters}
\]
5. **Calculate the area of the sidewalk:**
The area of the sidewalk is the difference between the area of the larger circle and the area of the flower bed.
\[
A_{\text{sidewalk}} = A_{\text{larger}} - A_{\text{flower bed}} = \pi \times 121 - \pi \times 64 = \pi (121 - 64) = \pi \times 57
\]
Therefore, the area of the sidewalk is:
\[
\boxed{\pi \times 57} \text{ square meters}
\]
#### Final Answer:
The area of the walk is \( \pi \times 57 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc0b40bc-1527-4d3e-b886-03bf08f02f26%2F1f54ca69-0656-466c-8377-0f756d4ddea7%2F7l6rx1v_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
A circular flower bed is 16 meters in diameter and has a circular sidewalk around it that is 3 meters wide. Find the area of the sidewalk in square meters.
#### Solution:
To find the area of the sidewalk, we first need to calculate the area of the larger circle (including the sidewalk) and the area of the flower bed (the smaller circle).
1. **Calculate the radius of the flower bed:**
The diameter of the flower bed is given as 16 meters. Therefore, the radius (r) is half of the diameter.
\[
r_{\text{flower bed}} = \frac{16}{2} = 8 \text{ meters}
\]
2. **Calculate the radius of the larger circle (flower bed + sidewalk):**
The sidewalk is 3 meters wide, so we add this width to the radius of the flower bed.
\[
r_{\text{total}} = r_{\text{flower bed}} + 3 = 8 + 3 = 11 \text{ meters}
\]
3. **Calculate the area of the larger circle:**
Using the formula for the area of a circle, \(A = \pi r^2\),
\[
A_{\text{larger}} = \pi \times 11^2 = \pi \times 121 \text{ square meters}
\]
4. **Calculate the area of the flower bed:**
Again, using the formula for the area of a circle,
\[
A_{\text{flower bed}} = \pi \times 8^2 = \pi \times 64 \text{ square meters}
\]
5. **Calculate the area of the sidewalk:**
The area of the sidewalk is the difference between the area of the larger circle and the area of the flower bed.
\[
A_{\text{sidewalk}} = A_{\text{larger}} - A_{\text{flower bed}} = \pi \times 121 - \pi \times 64 = \pi (121 - 64) = \pi \times 57
\]
Therefore, the area of the sidewalk is:
\[
\boxed{\pi \times 57} \text{ square meters}
\]
#### Final Answer:
The area of the walk is \( \pi \times 57 \)
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