Question
A cylinder and a cone are shown below.
A cylinder with height 12 inches and volume 2,512 inches cubed. A cone with height 12 inches and volume 1,256 inches cubed.
Which explains whether the bases of the cylinder and the cone have the same area?
The bases have the same area because the heights are the same.
The bases have the same area because the volume of the cone is One-half the volume of the cylinder.
The bases do not have the same area because the volumes are not the same.
The bases do not have the same area because the volume of the cylinder is not 3 times the volume of the cone, given the same heights.
A cylinder with height 12 inches and volume 2,512 inches cubed. A cone with height 12 inches and volume 1,256 inches cubed.
Which explains whether the bases of the cylinder and the cone have the same area?
The bases have the same area because the heights are the same.
The bases have the same area because the volume of the cone is One-half the volume of the cylinder.
The bases do not have the same area because the volumes are not the same.
The bases do not have the same area because the volume of the cylinder is not 3 times the volume of the cone, given the same heights.
Answers
Answered by
GPT-4o mini
To determine whether the bases of the cylinder and the cone have the same area, we can use the formula for the volumes of the cylinder and the cone. The volume formulas are as follows:
1. **Volume of a cylinder**: \( V = \pi r^2 h \)
2. **Volume of a cone**: \( V = \frac{1}{3} \pi r^2 h \)
Where \( r \) is the radius of the base, \( h \) is the height, and \( \pi \) is a constant (approximately 3.14).
Given:
- **Cylinder**: Height \( h_{cyl} = 12 \) inches, Volume \( V_{cyl} = 2,512 \) cubic inches
- **Cone**: Height \( h_{cone} = 12 \) inches, Volume \( V_{cone} = 1,256 \) cubic inches
From the volume of the cone, we see that the volume of the cone is one-third of the volume of the cylinder:
\[
V_{cone} = \frac{1}{3} V_{cyl}
\]
Since the heights of both the cylinder and the cone are equal, we can write the equations for their volumes in terms of their bases' areas (denoted as \( A \)):
1. For the cylinder:
\[
V_{cyl} = A_{cyl} \cdot h_{cyl} \quad \Rightarrow \quad A_{cyl} = \frac{V_{cyl}}{h_{cyl}} = \frac{2512}{12} = 209.33 \text{ square inches}
\]
2. For the cone:
\[
V_{cone} = \frac{1}{3} A_{cone} \cdot h_{cone} \quad \Rightarrow \quad A_{cone} = \frac{3V_{cone}}{h_{cone}} = \frac{3 \times 1256}{12} = 315.6 \text{ square inches}
\]
Since the areas calculated for the bases of the cylinder and the cone are not equal (209.33 ≠ 315.6), we conclude:
The bases do not have the same area because the volume of the cylinder is not three times the volume of the cone, given the same heights.
Thus, the correct explanation is:
**The bases do not have the same area because the volume of the cylinder is not 3 times the volume of the cone, given the same heights.**
1. **Volume of a cylinder**: \( V = \pi r^2 h \)
2. **Volume of a cone**: \( V = \frac{1}{3} \pi r^2 h \)
Where \( r \) is the radius of the base, \( h \) is the height, and \( \pi \) is a constant (approximately 3.14).
Given:
- **Cylinder**: Height \( h_{cyl} = 12 \) inches, Volume \( V_{cyl} = 2,512 \) cubic inches
- **Cone**: Height \( h_{cone} = 12 \) inches, Volume \( V_{cone} = 1,256 \) cubic inches
From the volume of the cone, we see that the volume of the cone is one-third of the volume of the cylinder:
\[
V_{cone} = \frac{1}{3} V_{cyl}
\]
Since the heights of both the cylinder and the cone are equal, we can write the equations for their volumes in terms of their bases' areas (denoted as \( A \)):
1. For the cylinder:
\[
V_{cyl} = A_{cyl} \cdot h_{cyl} \quad \Rightarrow \quad A_{cyl} = \frac{V_{cyl}}{h_{cyl}} = \frac{2512}{12} = 209.33 \text{ square inches}
\]
2. For the cone:
\[
V_{cone} = \frac{1}{3} A_{cone} \cdot h_{cone} \quad \Rightarrow \quad A_{cone} = \frac{3V_{cone}}{h_{cone}} = \frac{3 \times 1256}{12} = 315.6 \text{ square inches}
\]
Since the areas calculated for the bases of the cylinder and the cone are not equal (209.33 ≠ 315.6), we conclude:
The bases do not have the same area because the volume of the cylinder is not three times the volume of the cone, given the same heights.
Thus, the correct explanation is:
**The bases do not have the same area because the volume of the cylinder is not 3 times the volume of the cone, given the same heights.**
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