A cylinder of clay has a diameter of 12 cm and a height of 12 cm. If this clay is made into congruent cones with a radius of 3 cm and a height of 8 cm, how many cones could be made with all of the clay? If you used all the clay, you could make full cones.
A cylinder of clay has a diameter of 12 cm and a height of 12 cm. If this clay is made into congruent cones with a radius of 3 cm and a height of 8 cm, how many cones could be made with all of the clay? If you used all the clay, you could make full cones.
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
![**Problem Statement:**
A cylinder of clay has a diameter of 12 cm and a height of 12 cm. If this clay is made into congruent cones with a radius of 3 cm and a height of 8 cm, how many cones could be made with all of the clay?
If you used all the clay, you could make [ ] full cones.
**Detailed Explanation:**
To solve this problem, you need to understand the volume formulas for both a cylinder and a cone.
1. **Volume of a Cylinder**:
The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
For the given cylinder:
- Diameter = 12 cm, so radius \( r = \frac{12}{2} = 6 \) cm
- Height \( h = 12 \) cm
Substituting these values into the formula:
\[
V_{\text{cylinder}} = \pi \times 6^2 \times 12 = 432\pi \, \text{cubic centimeters}
\]
2. **Volume of a Cone**:
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
For each cone:
- Radius \( r = 3 \) cm
- Height \( h = 8 \) cm
Substituting these values into the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi \times 3^2 \times 8 = 24\pi \, \text{cubic centimeters}
\]
3. **Number of Cones**:
To find the number of cones that can be made from the cylinder, divide the volume of the cylinder by the volume of one cone:
\[
\text{Number of cones} = \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{432\pi}{24\pi} = 18
\]
Thus, if you](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e97dac8-8c85-49d8-8f7a-f33a844669bb%2F76a1cede-240b-4371-886f-416e67a50b33%2Fwdzjsn6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A cylinder of clay has a diameter of 12 cm and a height of 12 cm. If this clay is made into congruent cones with a radius of 3 cm and a height of 8 cm, how many cones could be made with all of the clay?
If you used all the clay, you could make [ ] full cones.
**Detailed Explanation:**
To solve this problem, you need to understand the volume formulas for both a cylinder and a cone.
1. **Volume of a Cylinder**:
The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
For the given cylinder:
- Diameter = 12 cm, so radius \( r = \frac{12}{2} = 6 \) cm
- Height \( h = 12 \) cm
Substituting these values into the formula:
\[
V_{\text{cylinder}} = \pi \times 6^2 \times 12 = 432\pi \, \text{cubic centimeters}
\]
2. **Volume of a Cone**:
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
For each cone:
- Radius \( r = 3 \) cm
- Height \( h = 8 \) cm
Substituting these values into the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi \times 3^2 \times 8 = 24\pi \, \text{cubic centimeters}
\]
3. **Number of Cones**:
To find the number of cones that can be made from the cylinder, divide the volume of the cylinder by the volume of one cone:
\[
\text{Number of cones} = \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{432\pi}{24\pi} = 18
\]
Thus, if you
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