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.
Question
Please find the volume of a sector of the following cylinder if the central angle is 65 degrees
![### Introduction to Cylinders
A cylinder is a three-dimensional geometric shape with two parallel bases that are usually circular. The line segments that join corresponding points on the two bases are parallel and of equal length. Here's an example of a cylinder with specific dimensions:
#### Diagram Explanation
In the diagram provided, we can observe the following dimensions of the cylinder:
1. **Radius of the Base (r)**: The radius of the base of the cylinder is indicated as 3 centimeters (cm). The radius is the distance from the center of the circle to any point on its circumference.
2. **Height of the Cylinder (h)**: The height of the cylinder is given as 10 centimeters (cm). The height is the distance between the two bases along the axis of the cylinder.
These dimensions define the size and shape of the cylinder. Understanding these measurements is crucial for calculating both the surface area and the volume of the cylinder.
### Calculating the Volume of a Cylinder
The volume \( V \) of a cylinder can be calculated using the formula:
\[ V = \pi r^2 h \]
Where:
- \( r \) is the radius of the base.
- \( h \) is the height of the cylinder.
- \( \pi \) (pi) is approximately equal to 3.14159.
### Calculating the Surface Area of a Cylinder
The surface area \( A \) of a cylinder is the total area of the outer surfaces and can be calculated using the formula:
\[ A = 2\pi r (r + h) \]
Where:
- \( r \) is the radius of the base.
- \( h \) is the height of the cylinder.
Understanding these formulas and the dimensions provided can help solve various problems related to cylinders in mathematical contexts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff488f205-bbfa-4c2f-9bd1-c5a845725e9f%2Fd78348af-f26f-4f2f-93ec-3248c9d8e455%2Fmvcak8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Introduction to Cylinders
A cylinder is a three-dimensional geometric shape with two parallel bases that are usually circular. The line segments that join corresponding points on the two bases are parallel and of equal length. Here's an example of a cylinder with specific dimensions:
#### Diagram Explanation
In the diagram provided, we can observe the following dimensions of the cylinder:
1. **Radius of the Base (r)**: The radius of the base of the cylinder is indicated as 3 centimeters (cm). The radius is the distance from the center of the circle to any point on its circumference.
2. **Height of the Cylinder (h)**: The height of the cylinder is given as 10 centimeters (cm). The height is the distance between the two bases along the axis of the cylinder.
These dimensions define the size and shape of the cylinder. Understanding these measurements is crucial for calculating both the surface area and the volume of the cylinder.
### Calculating the Volume of a Cylinder
The volume \( V \) of a cylinder can be calculated using the formula:
\[ V = \pi r^2 h \]
Where:
- \( r \) is the radius of the base.
- \( h \) is the height of the cylinder.
- \( \pi \) (pi) is approximately equal to 3.14159.
### Calculating the Surface Area of a Cylinder
The surface area \( A \) of a cylinder is the total area of the outer surfaces and can be calculated using the formula:
\[ A = 2\pi r (r + h) \]
Where:
- \( r \) is the radius of the base.
- \( h \) is the height of the cylinder.
Understanding these formulas and the dimensions provided can help solve various problems related to cylinders in mathematical contexts.
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