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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Concept explainers
Cylinders
A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.
Cones
A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is drawn by joining the apex to all points on the base, using segments, lines, or half-lines, provided that the apex and the base both are in different planes.
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![### Problem Statement
**4. A cylindrical hole was drilled through a cylinder. Find the volume. Give answer in terms of \( \pi \).**
### Diagram Description
The diagram illustrates a larger vertical cylinder with a smaller, concentric cylindrical hole drilled through it. The dimensions provided in the diagram are:
- The radius of the larger cylinder: 3 inches
- The radius of the hole (smaller cylinder): 1.5 inches
- The height of both cylinders: 15 inches
### Solution
To find the volume of the solid remaining after the hole is drilled, we need to calculate the volume of the larger cylinder and subtract the volume of the smaller cylinder (the hole).
1. **Volume of the larger cylinder**
\[ V_{\text{larger}} = \pi \times (3 \text{ in})^2 \times 15 \text{ in} \]
\[ V_{\text{larger}} = \pi \times 9 \text{ in}^2 \times 15 \text{ in} \]
\[ V_{\text{larger}} = 135\pi \text{ in}^3 \]
2. **Volume of the smaller cylinder (the hole)**
\[ V_{\text{smaller}} = \pi \times (1.5 \text{ in})^2 \times 15 \text{ in} \]
\[ V_{\text{smaller}} = \pi \times 2.25 \text{ in}^2 \times 15 \text{ in} \]
\[ V_{\text{smaller}} = 33.75\pi \text{ in}^3 \]
3. **Volume of the remaining solid**
\[ V_{\text{remaining}} = V_{\text{larger}} - V_{\text{smaller}} \]
\[ V_{\text{remaining}} = 135\pi \text{ in}^3 - 33.75\pi \text{ in}^3 \]
\[ V_{\text{remaining}} = 101.25\pi \text{ in}^3 \]
Therefore, the volume of the remaining solid is \( 101.25\pi \) cubic inches.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F35133500-1226-4a9c-afb4-40ee5a709984%2Fba3c0d40-0026-458f-bab5-ce1df93891d2%2Fxbpff7s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**4. A cylindrical hole was drilled through a cylinder. Find the volume. Give answer in terms of \( \pi \).**
### Diagram Description
The diagram illustrates a larger vertical cylinder with a smaller, concentric cylindrical hole drilled through it. The dimensions provided in the diagram are:
- The radius of the larger cylinder: 3 inches
- The radius of the hole (smaller cylinder): 1.5 inches
- The height of both cylinders: 15 inches
### Solution
To find the volume of the solid remaining after the hole is drilled, we need to calculate the volume of the larger cylinder and subtract the volume of the smaller cylinder (the hole).
1. **Volume of the larger cylinder**
\[ V_{\text{larger}} = \pi \times (3 \text{ in})^2 \times 15 \text{ in} \]
\[ V_{\text{larger}} = \pi \times 9 \text{ in}^2 \times 15 \text{ in} \]
\[ V_{\text{larger}} = 135\pi \text{ in}^3 \]
2. **Volume of the smaller cylinder (the hole)**
\[ V_{\text{smaller}} = \pi \times (1.5 \text{ in})^2 \times 15 \text{ in} \]
\[ V_{\text{smaller}} = \pi \times 2.25 \text{ in}^2 \times 15 \text{ in} \]
\[ V_{\text{smaller}} = 33.75\pi \text{ in}^3 \]
3. **Volume of the remaining solid**
\[ V_{\text{remaining}} = V_{\text{larger}} - V_{\text{smaller}} \]
\[ V_{\text{remaining}} = 135\pi \text{ in}^3 - 33.75\pi \text{ in}^3 \]
\[ V_{\text{remaining}} = 101.25\pi \text{ in}^3 \]
Therefore, the volume of the remaining solid is \( 101.25\pi \) cubic inches.
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