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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Question
![### Problem Statement
**Objective:**
Find the height \( h \) of a right circular cone with the following given measurements:
- The diameter of the base \( d \) measures 9.6 cm
- The slant height \( l \) is 5.2 m
### Explanation of Diagrams
#### Diagram of the Cone:
There is an illustration of a right circular cone with the following labels:
- The slant height \( l \) is labeled on the side of the cone.
- The height \( h \) is labeled as the vertical distance from the base to the apex of the cone.
- The diameter \( d \) is labeled across the circular base of the cone.
#### Conversion of Units:
The diameter \( d \) is provided in centimeters and must be converted to meters for consistency in calculations.
\[ d = 9.6 \, \text{cm} = 0.096 \, \text{m} \]
### Formula Involved
To find the height \( h \), we use the Pythagorean theorem in the right triangle formed by the radius \( r \), slant height \( l \), and height \( h \) of the cone.
The relationship is given by:
\[ r^2 + h^2 = l^2 \]
Where \( r \) is the radius of the base:
\[ r = \frac{d}{2} \]
So, substituting the value of \( r \) and \( l \):
\[ r = \frac{0.096}{2} = 0.048 \, \text{m} \]
\[ (0.048)^2 + h^2 = (5.2)^2 \]
Which simplifies to:
\[ 0.002304 + h^2 = 27.04 \]
Solving for \( h \):
\[ h^2 = 27.04 - 0.002304 \]
\[ h^2 = 27.037696 \]
\[ h = \sqrt{27.037696} \]
\[ h \approx 5.2 \, \text{m} \]
Therefore, the height \( h \) of the right circular cone is approximately \( 5.2 \, \text{m} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55a87db1-34dd-4361-bb49-0fd4dde658a5%2F325a19b8-67c1-471b-92e9-dcd8f21cf103%2Fi0vsu0u.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Objective:**
Find the height \( h \) of a right circular cone with the following given measurements:
- The diameter of the base \( d \) measures 9.6 cm
- The slant height \( l \) is 5.2 m
### Explanation of Diagrams
#### Diagram of the Cone:
There is an illustration of a right circular cone with the following labels:
- The slant height \( l \) is labeled on the side of the cone.
- The height \( h \) is labeled as the vertical distance from the base to the apex of the cone.
- The diameter \( d \) is labeled across the circular base of the cone.
#### Conversion of Units:
The diameter \( d \) is provided in centimeters and must be converted to meters for consistency in calculations.
\[ d = 9.6 \, \text{cm} = 0.096 \, \text{m} \]
### Formula Involved
To find the height \( h \), we use the Pythagorean theorem in the right triangle formed by the radius \( r \), slant height \( l \), and height \( h \) of the cone.
The relationship is given by:
\[ r^2 + h^2 = l^2 \]
Where \( r \) is the radius of the base:
\[ r = \frac{d}{2} \]
So, substituting the value of \( r \) and \( l \):
\[ r = \frac{0.096}{2} = 0.048 \, \text{m} \]
\[ (0.048)^2 + h^2 = (5.2)^2 \]
Which simplifies to:
\[ 0.002304 + h^2 = 27.04 \]
Solving for \( h \):
\[ h^2 = 27.04 - 0.002304 \]
\[ h^2 = 27.037696 \]
\[ h = \sqrt{27.037696} \]
\[ h \approx 5.2 \, \text{m} \]
Therefore, the height \( h \) of the right circular cone is approximately \( 5.2 \, \text{m} \).
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