Question
A cone and a cylinder have the same base and height, as shown below.
A cone is inside of a cylinder with same base and height.
The volume of the cylinder is 84 pi inches squared What can be concluded about the volume of the cone? Check all that apply.
The volume of the cone is One-third the volume of the cylinder.
The volume of the cylinder is 2 times the volume of the cone.
The volume of the cone is less than the volume of the cylinder.
The expression StartFraction 84 pi Over 3 EndFraction finds the volume of the cone.
The expression mr023-5.jpg finds the volume of the cone.
A cone is inside of a cylinder with same base and height.
The volume of the cylinder is 84 pi inches squared What can be concluded about the volume of the cone? Check all that apply.
The volume of the cone is One-third the volume of the cylinder.
The volume of the cylinder is 2 times the volume of the cone.
The volume of the cone is less than the volume of the cylinder.
The expression StartFraction 84 pi Over 3 EndFraction finds the volume of the cone.
The expression mr023-5.jpg finds the volume of the cone.
Answers
Answered by
GPT-4o mini
To analyze the problem, let's start by recalling the formulas for the volumes of a cylinder and a cone.
1. **Volume of a cylinder**:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
2. **Volume of a cone**:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
Given that:
- The volume of the cylinder is \( 84\pi \) cubic inches.
We can conclude:
1. **The volume of the cone is one-third the volume of the cylinder.**
This is true; the volume of the cone is calculated to be:
\[
V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3} \times 84\pi = 28\pi
\]
2. **The volume of the cylinder is 2 times the volume of the cone.**
This is false. The volume of the cylinder is \( 3 \times \) the volume of the cone.
3. **The volume of the cone is less than the volume of the cylinder.**
This is true because \( V_{\text{cone}} = 28\pi \) and \( V_{\text{cylinder}} = 84\pi \).
4. **The expression \(\frac{84\pi}{3}\) finds the volume of the cone.**
This is true. As calculated previously, \( \frac{84\pi}{3} = 28\pi\), which is the volume of the cone.
5. **The expression \( \text{mr023-5.jpg} \) finds the volume of the cone.**
Without seeing the actual expression in that image, I cannot confirm this statement. Please check the expression in the image to see if it represents the volume of a cone.
In conclusion, the following apply:
- The volume of the cone is one-third the volume of the cylinder.
- The volume of the cone is less than the volume of the cylinder.
- The expression \(\frac{84\pi}{3}\) finds the volume of the cone.
1. **Volume of a cylinder**:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
2. **Volume of a cone**:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
Given that:
- The volume of the cylinder is \( 84\pi \) cubic inches.
We can conclude:
1. **The volume of the cone is one-third the volume of the cylinder.**
This is true; the volume of the cone is calculated to be:
\[
V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3} \times 84\pi = 28\pi
\]
2. **The volume of the cylinder is 2 times the volume of the cone.**
This is false. The volume of the cylinder is \( 3 \times \) the volume of the cone.
3. **The volume of the cone is less than the volume of the cylinder.**
This is true because \( V_{\text{cone}} = 28\pi \) and \( V_{\text{cylinder}} = 84\pi \).
4. **The expression \(\frac{84\pi}{3}\) finds the volume of the cone.**
This is true. As calculated previously, \( \frac{84\pi}{3} = 28\pi\), which is the volume of the cone.
5. **The expression \( \text{mr023-5.jpg} \) finds the volume of the cone.**
Without seeing the actual expression in that image, I cannot confirm this statement. Please check the expression in the image to see if it represents the volume of a cone.
In conclusion, the following apply:
- The volume of the cone is one-third the volume of the cylinder.
- The volume of the cone is less than the volume of the cylinder.
- The expression \(\frac{84\pi}{3}\) finds the volume of the cone.