Question
A sphere and a cylinder have the same radius and height. The volume of the cylinder is 21 meters cubed.
A sphere with height h and radius r. A cylinder with height h and radius r.
What is the volume of the sphere?
6 meters cubed
7 meters cubed
14 meters cubed
28 meters cubed
A sphere with height h and radius r. A cylinder with height h and radius r.
What is the volume of the sphere?
6 meters cubed
7 meters cubed
14 meters cubed
28 meters cubed
Answers
GPT-4o mini
The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
Since we know the volume of the cylinder is 21 cubic meters, we can use this information to find the relationship between the height \( h \) and the radius \( r \):
\[
\pi r^2 h = 21
\]
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
To find the volume of the sphere, we need to express it in terms of the known volume of the cylinder.
Since both the sphere and the cylinder have the same radius \( r \) and height \( h \), we can rearrange the cylinder's volume equation to find \( h \):
\[
h = \frac{21}{\pi r^2}
\]
Now we can substitute \( r \) into the sphere's volume formula. However, we can also find a relationship between their volumes directly:
1. The volume of the cylinder is given as 21 cubic meters.
2. The volume of a sphere with the same radius as the radius of the cylinder is \( \frac{2}{3} \) of the cylinder's volume.
This relationship gives us:
\[
V_{\text{sphere}} = \frac{2}{3} V_{\text{cylinder}} = \frac{2}{3} \times 21 = 14 \, \text{cubic meters}
\]
So, the volume of the sphere is \( 14 \) cubic meters. The correct answer is:
**14 meters cubed**.
\[
V = \pi r^2 h
\]
Since we know the volume of the cylinder is 21 cubic meters, we can use this information to find the relationship between the height \( h \) and the radius \( r \):
\[
\pi r^2 h = 21
\]
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
To find the volume of the sphere, we need to express it in terms of the known volume of the cylinder.
Since both the sphere and the cylinder have the same radius \( r \) and height \( h \), we can rearrange the cylinder's volume equation to find \( h \):
\[
h = \frac{21}{\pi r^2}
\]
Now we can substitute \( r \) into the sphere's volume formula. However, we can also find a relationship between their volumes directly:
1. The volume of the cylinder is given as 21 cubic meters.
2. The volume of a sphere with the same radius as the radius of the cylinder is \( \frac{2}{3} \) of the cylinder's volume.
This relationship gives us:
\[
V_{\text{sphere}} = \frac{2}{3} V_{\text{cylinder}} = \frac{2}{3} \times 21 = 14 \, \text{cubic meters}
\]
So, the volume of the sphere is \( 14 \) cubic meters. The correct answer is:
**14 meters cubed**.