To find the volume of the sphere, let’s start by understanding the relationship between the volumes of the cylinder and the sphere given their geometrical properties.
1. Volume of a Cylinder Formula:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
Given:
[tex]\[
V_{\text{cylinder}} = 21 \text{ m}^3
\][/tex]
2. Volume of a Sphere Formula:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\][/tex]
Since the height of the cylinder [tex]\( h \)[/tex] equals the diameter of the sphere, which is [tex]\( 2r \)[/tex], we can link the cylinder's volume to the sphere's radius:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3
\][/tex]
Given:
[tex]\[
21 = 2\pi r^3
\][/tex]
Now, we solve for [tex]\( r^3 \)[/tex]:
[tex]\[
r^3 = \frac{21}{2\pi}
\][/tex]
Next, we use the volume formula for the sphere:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\][/tex]
Substitute [tex]\( r^3 \)[/tex] from the previous step:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{21}{2\pi} \right)
\][/tex]
Simplify:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \cdot \frac{21}{2}
\][/tex]
[tex]\[
V_{\text{sphere}} = \frac{4 \cdot 21}{3 \cdot 2} = \frac{84}{6} = 14
\][/tex]
Therefore, the volume of the sphere is:
[tex]\[
14 \text{ m}^3
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{14 \text{ m}^3}
\][/tex]
