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
![**Finding the Volume of a Composite Shape**
To solve this problem:
1. Calculate the volume of the cylinder.
2. Calculate the volume of the hemisphere.
3. Add both volumes to get the total volume.
**Step 1: Volume of the Cylinder**
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where:
- \( r \) is the radius.
- \( h \) is the height.
For this cylinder:
- Radius \( r = 8 \) ft (16 ft diameter divided by 2).
- Height \( h = 24 \) ft.
\[ V_{\text{cylinder}} = \pi \times (8)^2 \times 24 \]
\[ V_{\text{cylinder}} = \pi \times 64 \times 24 \]
\[ V_{\text{cylinder}} = \pi \times 1536 \]
\[ V_{\text{cylinder}} \approx 4821.4 \text{ cubic feet} \]
**Step 2: Volume of the Hemisphere**
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3}\pi r^3 \]
A hemisphere is half of a sphere, so the volume \( V \) of a hemisphere is:
\[ V = \frac{1}{2} \times \frac{4}{3}\pi r^3 \]
For this hemisphere:
- Radius \( r = 8 \) ft.
\[ V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \pi \times (8)^3 \]
\[ V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \pi \times 512 \]
\[ V_{text{hemisphere}} = \frac{2}{3} \pi \times 512 \]
\[ V_{\text{hemisphere}} = \frac{1024}{3} \pi \]
\[ V_{\text{hemisphere}} \approx 1072.8 \text{ cubic feet} \]
**Step 3: Total Volume of the Composite Shape**
\[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} \]
\[ V_{\text{total}} \approx 4821](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ba5e651-ce33-4122-bea4-14db54826e08%2Ff84faadd-cf2f-4331-86e3-20aa2a15d208%2Fzuqii1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Volume of a Composite Shape**
To solve this problem:
1. Calculate the volume of the cylinder.
2. Calculate the volume of the hemisphere.
3. Add both volumes to get the total volume.
**Step 1: Volume of the Cylinder**
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where:
- \( r \) is the radius.
- \( h \) is the height.
For this cylinder:
- Radius \( r = 8 \) ft (16 ft diameter divided by 2).
- Height \( h = 24 \) ft.
\[ V_{\text{cylinder}} = \pi \times (8)^2 \times 24 \]
\[ V_{\text{cylinder}} = \pi \times 64 \times 24 \]
\[ V_{\text{cylinder}} = \pi \times 1536 \]
\[ V_{\text{cylinder}} \approx 4821.4 \text{ cubic feet} \]
**Step 2: Volume of the Hemisphere**
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3}\pi r^3 \]
A hemisphere is half of a sphere, so the volume \( V \) of a hemisphere is:
\[ V = \frac{1}{2} \times \frac{4}{3}\pi r^3 \]
For this hemisphere:
- Radius \( r = 8 \) ft.
\[ V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \pi \times (8)^3 \]
\[ V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3} \pi \times 512 \]
\[ V_{text{hemisphere}} = \frac{2}{3} \pi \times 512 \]
\[ V_{\text{hemisphere}} = \frac{1024}{3} \pi \]
\[ V_{\text{hemisphere}} \approx 1072.8 \text{ cubic feet} \]
**Step 3: Total Volume of the Composite Shape**
\[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{hemisphere}} \]
\[ V_{\text{total}} \approx 4821
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