5. Describe a hemisphere of this picture and explain how it was formed. If given the radius of the globe, how would you find the volume of the hemisphere? 国
5. Describe a hemisphere of this picture and explain how it was formed. If given the radius of the globe, how would you find the volume of the hemisphere? 国
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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![## Understanding Hemispheres and Volume Calculation
### Question:
Describe a hemisphere of this picture and explain how it was formed. If given the radius of the globe, how would you find the volume of the hemisphere?
### Answer:
A hemisphere is half of a sphere, resulting from cutting a sphere along a plane passing through its center. In the picture, the globe represents a sphere of the Earth, which can be conceptually divided into two equal halves or hemispheres — the Northern Hemisphere and the Southern Hemisphere, or alternatively, the Eastern and Western Hemispheres.
### Formation of Hemispheres:
Hemispheres are formed by the intersection of the sphere with a plane that cuts through the center of the sphere. This plane effectively halves the sphere into two equal parts. In the case of Earth, the Equator divides the Northern and Southern Hemispheres, while the Prime Meridian and the 180th meridian divide the Eastern and Western Hemispheres.
### Volume Calculation:
To find the volume of a hemisphere given the radius \( r \) of the globe, you can use the volume formula for a sphere and then divide it by two, since a hemisphere is half of a sphere.
1. **Volume of a Sphere:**
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
2. **Volume of a Hemisphere:**
To find the volume of a hemisphere, divide the volume of the sphere by 2:
\[
V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]
By using this formula, you can easily calculate the volume of any hemisphere if the radius \( r \) of the sphere is known.
### Graph/Diagram Explanation:
The image includes a globe which serves as a three-dimensional model of the Earth. This visual aid helps to conceptualize the notion of hemispheres. The globe is mounted on a stand, with a semi-circular frame that likely represents the Earth's rotation axis. This model is useful for physical demonstrations of global geography and the division of the Earth's hemispheres.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e4c2504-a404-4a5a-a28b-2890154f0d9e%2F93039e63-75ca-440b-b8f0-648e06622f14%2Fw49whl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Understanding Hemispheres and Volume Calculation
### Question:
Describe a hemisphere of this picture and explain how it was formed. If given the radius of the globe, how would you find the volume of the hemisphere?
### Answer:
A hemisphere is half of a sphere, resulting from cutting a sphere along a plane passing through its center. In the picture, the globe represents a sphere of the Earth, which can be conceptually divided into two equal halves or hemispheres — the Northern Hemisphere and the Southern Hemisphere, or alternatively, the Eastern and Western Hemispheres.
### Formation of Hemispheres:
Hemispheres are formed by the intersection of the sphere with a plane that cuts through the center of the sphere. This plane effectively halves the sphere into two equal parts. In the case of Earth, the Equator divides the Northern and Southern Hemispheres, while the Prime Meridian and the 180th meridian divide the Eastern and Western Hemispheres.
### Volume Calculation:
To find the volume of a hemisphere given the radius \( r \) of the globe, you can use the volume formula for a sphere and then divide it by two, since a hemisphere is half of a sphere.
1. **Volume of a Sphere:**
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
2. **Volume of a Hemisphere:**
To find the volume of a hemisphere, divide the volume of the sphere by 2:
\[
V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]
By using this formula, you can easily calculate the volume of any hemisphere if the radius \( r \) of the sphere is known.
### Graph/Diagram Explanation:
The image includes a globe which serves as a three-dimensional model of the Earth. This visual aid helps to conceptualize the notion of hemispheres. The globe is mounted on a stand, with a semi-circular frame that likely represents the Earth's rotation axis. This model is useful for physical demonstrations of global geography and the division of the Earth's hemispheres.
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