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
Topic Video
Question
![### Understanding Cylinders and Cones with Congruent Bases and Equal Heights
In this educational overview, we will explore the properties of a cylinder and a cone that share congruent bases and equal heights.
#### Visual Representation:
The provided image illustrates a cylinder and a cone side by side. Both shapes have a height of 3 meters and a base area of \(12 \, \text{m}^2\).
1. **Cylinder:**
- The cylinder is depicted with a circular base and straight vertical sides.
- The height of the cylinder is labeled as 3 meters.
- The area of the base is given as \(12 \, \text{m}^2\).
2. **Cone:**
- The cone has a circular base, indicated as congruent to the base of the cylinder.
- The vertical height from the base to the apex of the cone is also 3 meters.
- The area of the base of the cone is \(12 \, \text{m}^2\), matching the base area of the cylinder.
#### Key Concepts:
**Volume Calculation:**
- **Cylinder Volume Formula:**
\[
V = \pi r^2 h
\]
Where:
- \( V \) is the volume
- \( r \) is the radius of the base
- \( h \) is the height of the cylinder
- **Cone Volume Formula:**
\[
V = \frac{1}{3} \pi r^2 h
\]
Where:
- \( V \) is the volume
- \( r \) is the radius of the base
- \( h \) is the height of the cone
Given that both shapes have the same base area \( (12 \, \text{m}^2) \) and height \( (3 \, \text{m}) \), one can infer how their volumes would compare. Specifically:
- The volume of the cone will be one-third of the volume of the cylinder because the height multiplied by the area of the base is divided by 3 in the formula for the cone.
**Base Area and Height:**
Understanding that the cylinder and cone have congruent bases and equal heights is crucial. This congruence implies that regardless of the difference in their three-dimensional structures, the circular bases are identical (same radius and area).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6373081f-17d9-417b-b535-bd8706276caa%2F55e147ea-c548-4651-a2e2-6c411c055390%2Fjsmq30c_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Cylinders and Cones with Congruent Bases and Equal Heights
In this educational overview, we will explore the properties of a cylinder and a cone that share congruent bases and equal heights.
#### Visual Representation:
The provided image illustrates a cylinder and a cone side by side. Both shapes have a height of 3 meters and a base area of \(12 \, \text{m}^2\).
1. **Cylinder:**
- The cylinder is depicted with a circular base and straight vertical sides.
- The height of the cylinder is labeled as 3 meters.
- The area of the base is given as \(12 \, \text{m}^2\).
2. **Cone:**
- The cone has a circular base, indicated as congruent to the base of the cylinder.
- The vertical height from the base to the apex of the cone is also 3 meters.
- The area of the base of the cone is \(12 \, \text{m}^2\), matching the base area of the cylinder.
#### Key Concepts:
**Volume Calculation:**
- **Cylinder Volume Formula:**
\[
V = \pi r^2 h
\]
Where:
- \( V \) is the volume
- \( r \) is the radius of the base
- \( h \) is the height of the cylinder
- **Cone Volume Formula:**
\[
V = \frac{1}{3} \pi r^2 h
\]
Where:
- \( V \) is the volume
- \( r \) is the radius of the base
- \( h \) is the height of the cone
Given that both shapes have the same base area \( (12 \, \text{m}^2) \) and height \( (3 \, \text{m}) \), one can infer how their volumes would compare. Specifically:
- The volume of the cone will be one-third of the volume of the cylinder because the height multiplied by the area of the base is divided by 3 in the formula for the cone.
**Base Area and Height:**
Understanding that the cylinder and cone have congruent bases and equal heights is crucial. This congruence implies that regardless of the difference in their three-dimensional structures, the circular bases are identical (same radius and area).
Transcribed Image Text:**Volume Calculation Problem Set**
**Complete the following:**
(a) Volume of the cylinder: \(\mathbf{\square}\) \(\text{m}^3\)
(b) Volume of the cone: \(\mathbf{\square}\) \(\text{m}^3\)
(c) Volume of the cone = \(\mathbf{\square}\) × Volume of the cylinder
- ☐ This equation is true only for the cylinder and cone shown above.
- ☐ This equation is true for all cylinders and cones.
- ☐ This equation is true for all cylinders and cones with congruent bases and equal heights.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning