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
how do i find surface area
![### Surface Area Calculation of a Sphere
**Given Data:**
- The circumference of the great circle of the sphere is \( 49\pi \) cm.
### Diagram Description:
The image illustrates a sphere. A circle is drawn within the sphere with a dotted line to represent the great circle. The circumference of the great circle is marked as \( 49\pi \) cm. Below the diagram, there is a placeholder to enter the final answer, designated as “_______ square cm.”
### Step-by-Step Solution:
1. **Understand the Circumference Formula:**
The circumference \( C \) of a circle is calculated using the formula:
\[
C = 2\pi r
\]
where \( r \) is the radius of the circle.
2. **Determine the Radius:**
Given that the circumference \( C \) is \( 49\pi \) cm, we can set up the equation:
\[
49\pi = 2\pi r
\]
To solve for \( r \), divide both sides by \( 2\pi \):
\[
r = \frac{49\pi}{2\pi} = \frac{49}{2} = 24.5 \text{ cm}
\]
3. **Calculate the Surface Area:**
The surface area \( A \) of a sphere is given by the formula:
\[
A = 4\pi r^2
\]
Substituting the radius \( r = 24.5 \) cm, we find:
\[
A = 4\pi (24.5)^2
\]
Calculate \( 24.5^2 \):
\[
24.5 \times 24.5 = 600.25
\]
Substitute back into the original formula:
\[
A = 4\pi \times 600.25 = 2401\pi \text{ square cm}
\]
4. **Final Answer:**
The surface area of the sphere is \( 2401\pi \) square cm.
### Conclusion:
Using the given circumference of the great circle, we determined the sphere's radius and surface area. The surface area is presented in terms of \( \pi \) as \( 2401\pi \) square cm.
**Answer](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0952b748-9a71-48e2-8bc6-ad391431fcc1%2F20c0aee6-db3e-4f94-a155-9f7bf23f3639%2F8j5h6z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Surface Area Calculation of a Sphere
**Given Data:**
- The circumference of the great circle of the sphere is \( 49\pi \) cm.
### Diagram Description:
The image illustrates a sphere. A circle is drawn within the sphere with a dotted line to represent the great circle. The circumference of the great circle is marked as \( 49\pi \) cm. Below the diagram, there is a placeholder to enter the final answer, designated as “_______ square cm.”
### Step-by-Step Solution:
1. **Understand the Circumference Formula:**
The circumference \( C \) of a circle is calculated using the formula:
\[
C = 2\pi r
\]
where \( r \) is the radius of the circle.
2. **Determine the Radius:**
Given that the circumference \( C \) is \( 49\pi \) cm, we can set up the equation:
\[
49\pi = 2\pi r
\]
To solve for \( r \), divide both sides by \( 2\pi \):
\[
r = \frac{49\pi}{2\pi} = \frac{49}{2} = 24.5 \text{ cm}
\]
3. **Calculate the Surface Area:**
The surface area \( A \) of a sphere is given by the formula:
\[
A = 4\pi r^2
\]
Substituting the radius \( r = 24.5 \) cm, we find:
\[
A = 4\pi (24.5)^2
\]
Calculate \( 24.5^2 \):
\[
24.5 \times 24.5 = 600.25
\]
Substitute back into the original formula:
\[
A = 4\pi \times 600.25 = 2401\pi \text{ square cm}
\]
4. **Final Answer:**
The surface area of the sphere is \( 2401\pi \) square cm.
### Conclusion:
Using the given circumference of the great circle, we determined the sphere's radius and surface area. The surface area is presented in terms of \( \pi \) as \( 2401\pi \) square cm.
**Answer
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