b. Please solve the following question. You need to show ALL your work that leads to the final answer. Jeremy designed a capsule container with dimensions shown below. Please help him find the volume of the container, to the nearest hundredth of inches. 2 in. 8 in.
b. Please solve the following question. You need to show ALL your work that leads to the final answer. Jeremy designed a capsule container with dimensions shown below. Please help him find the volume of the container, to the nearest hundredth of inches. 2 in. 8 in.
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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![### Problem Statement
**b. Please solve the following question. You need to show ALL your work that leads to the final answer.**
**Jeremy designed a capsule container with dimensions shown below. Please help him find the volume of the container, to the nearest hundredth of inches.**
#### Diagram Description
The diagram illustrates a capsule-shaped container composed of a cylinder with hemispherical ends. The dimensions given are:
- The radius of the hemispheres and the cylinder is 2 inches.
- The length of the cylindrical section of the capsule is 8 inches.
### Solution Explanation
To determine the volume of the capsule container, it is necessary to calculate the volume of the cylindrical section and the volume of the hemispheres:
#### Volume of the Cylinder
\[ \text{Volume of Cylinder} (V_{\text{cylinder}}) = \pi r^2 h \]
Where:
- \( r \) is the radius of the cylinder (2 inches),
- \( h \) is the height (length) of the cylinder (8 inches).
#### Volume of the Hemispheres
Since there are two hemispherical ends, their combined volume will be equivalent to the volume of a single sphere:
\[ \text{Volume of Sphere} (V_{\text{sphere}}) = \frac{4}{3} \pi r^3 \]
Thus, the total volume of the capsule is the sum of the volume of the cylindrical part and the volume of the sphere:
\[ \text{Total Volume} = V_{\text{cylinder}} + V_{\text{sphere}} \]
### Calculation Steps
1. Calculate the volume of the cylinder:
\[ V_{\text{cylinder}} = \pi (2^2) \times 8 \]
\[ V_{\text{cylinder}} = \pi \times 4 \times 8 \]
\[ V_{\text{cylinder}} = 32\pi \]
2. Calculate the volume of the sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (2^3) \]
\[ V_{\text{sphere}} = \frac{4}{3} \pi \times 8 \]
\[ V_{\text{sphere}} = \frac{32}{3} \pi \]
3. Add both volumes to find total volume:
\[ \text{Total Volume} = 32\pi + \frac{32}{3}\pi \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5632a8be-345b-47d3-aa13-88c8cf1cba30%2Ff7939d24-7a60-4a1b-a755-cb34652cebf8%2Fqshyuxd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**b. Please solve the following question. You need to show ALL your work that leads to the final answer.**
**Jeremy designed a capsule container with dimensions shown below. Please help him find the volume of the container, to the nearest hundredth of inches.**
#### Diagram Description
The diagram illustrates a capsule-shaped container composed of a cylinder with hemispherical ends. The dimensions given are:
- The radius of the hemispheres and the cylinder is 2 inches.
- The length of the cylindrical section of the capsule is 8 inches.
### Solution Explanation
To determine the volume of the capsule container, it is necessary to calculate the volume of the cylindrical section and the volume of the hemispheres:
#### Volume of the Cylinder
\[ \text{Volume of Cylinder} (V_{\text{cylinder}}) = \pi r^2 h \]
Where:
- \( r \) is the radius of the cylinder (2 inches),
- \( h \) is the height (length) of the cylinder (8 inches).
#### Volume of the Hemispheres
Since there are two hemispherical ends, their combined volume will be equivalent to the volume of a single sphere:
\[ \text{Volume of Sphere} (V_{\text{sphere}}) = \frac{4}{3} \pi r^3 \]
Thus, the total volume of the capsule is the sum of the volume of the cylindrical part and the volume of the sphere:
\[ \text{Total Volume} = V_{\text{cylinder}} + V_{\text{sphere}} \]
### Calculation Steps
1. Calculate the volume of the cylinder:
\[ V_{\text{cylinder}} = \pi (2^2) \times 8 \]
\[ V_{\text{cylinder}} = \pi \times 4 \times 8 \]
\[ V_{\text{cylinder}} = 32\pi \]
2. Calculate the volume of the sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (2^3) \]
\[ V_{\text{sphere}} = \frac{4}{3} \pi \times 8 \]
\[ V_{\text{sphere}} = \frac{32}{3} \pi \]
3. Add both volumes to find total volume:
\[ \text{Total Volume} = 32\pi + \frac{32}{3}\pi \
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