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
Find the volume of the figure provided
![**Volume Calculation: Composite Figure**
### Problem Statement
Determine the volume of the composite figure. Round your answer to the nearest hundredth.
### Diagram Description
The composite figure consists of a cone placed on top of a cylinder.
- The cone has a height of 4 inches and a base radius of 2 inches.
- The cylinder has a height of 5 inches and also has a radius of 2 inches.
### Steps to Solve
#### 1. Calculate the Volume of the Cylinder
The formula for the volume of a cylinder is given by:
\[
V_{cylinder} = \pi r^2 h
\]
Given:
- \( r = 2 \; \text{inches} \)
- \( h = 5 \; \text{inches} \)
\[
V_{cylinder} = \pi (2)^2 (5) = 20\pi
\]
#### 2. Calculate the Volume of the Cone
The formula for the volume of a cone is given by:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Given:
- \( r = 2 \; \text{inches} \)
- \( h = 4 \; \text{inches} \)
\[
V_{cone} = \frac{1}{3} \pi (2)^2 (4) = \frac{16\pi}{3}
\]
#### 3. Sum the Volumes
To find the total volume \( V_{total} \):
\[
V_{total} = V_{cylinder} + V_{cone}
\]
Substitute the values:
\[
V_{total} = 20\pi + \frac{16\pi}{3}
\]
Convert to a common denominator to sum:
\[
V_{total} = \frac{60\pi}{3} + \frac{16\pi}{3} = \frac{76\pi}{3}
\]
#### 4. Calculate the Numerical Value
\[
V_{total} \approx \frac{76 \times 3.14159}{3} \approx 79.58 \; \text{cubic inches}
\]
### Final Answer
The volume of the composite figure, rounded to the nearest hundredth, is approximately \( 79.58 \; \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95e9d242-cb37-4bc8-a74f-f7cca1efbc15%2Ff57da5c3-7769-489a-816b-13840480adea%2Fa80najs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Volume Calculation: Composite Figure**
### Problem Statement
Determine the volume of the composite figure. Round your answer to the nearest hundredth.
### Diagram Description
The composite figure consists of a cone placed on top of a cylinder.
- The cone has a height of 4 inches and a base radius of 2 inches.
- The cylinder has a height of 5 inches and also has a radius of 2 inches.
### Steps to Solve
#### 1. Calculate the Volume of the Cylinder
The formula for the volume of a cylinder is given by:
\[
V_{cylinder} = \pi r^2 h
\]
Given:
- \( r = 2 \; \text{inches} \)
- \( h = 5 \; \text{inches} \)
\[
V_{cylinder} = \pi (2)^2 (5) = 20\pi
\]
#### 2. Calculate the Volume of the Cone
The formula for the volume of a cone is given by:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Given:
- \( r = 2 \; \text{inches} \)
- \( h = 4 \; \text{inches} \)
\[
V_{cone} = \frac{1}{3} \pi (2)^2 (4) = \frac{16\pi}{3}
\]
#### 3. Sum the Volumes
To find the total volume \( V_{total} \):
\[
V_{total} = V_{cylinder} + V_{cone}
\]
Substitute the values:
\[
V_{total} = 20\pi + \frac{16\pi}{3}
\]
Convert to a common denominator to sum:
\[
V_{total} = \frac{60\pi}{3} + \frac{16\pi}{3} = \frac{76\pi}{3}
\]
#### 4. Calculate the Numerical Value
\[
V_{total} \approx \frac{76 \times 3.14159}{3} \approx 79.58 \; \text{cubic inches}
\]
### Final Answer
The volume of the composite figure, rounded to the nearest hundredth, is approximately \( 79.58 \; \
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