To determine the volume of the sphere given that it has the same radius and height as a cylinder with a volume of [tex]\(48 \, \text{cm}^3\)[/tex], follow these steps:
1. Volume of the Cylinder:
The volume of a cylinder is calculated using the formula:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
Here, [tex]\(r\)[/tex] is the radius and [tex]\(h\)[/tex] is the height.
2. Interrelation of Dimensions:
Since the cylinder and the sphere have the same radius [tex]\(r\)[/tex] and height, for the sphere the diameter is equal to the cylinder's height [tex]\(h\)[/tex], which means:
[tex]\[
h = 2r
\][/tex]
3. Simplification for Sphere's Volume:
The volume [tex]\(V_{\text{sphere}}\)[/tex] of a sphere is given by:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\][/tex]
4. Relating Volumes:
From the height [tex]\( h = 2r \)[/tex] and the volume of the cylinder [tex]\( V_{\text{cylinder}} = \pi r^2 h \)[/tex]:
[tex]\[
V_{\text{cylinder}} = \pi r^2 \cdot 2r = 2 \pi r^3
\][/tex]
Therefore:
[tex]\[
2 \pi r^3 = 48 \implies r^3 = \frac{48}{2 \pi}
\][/tex]
5. Volume of the Sphere:
The volume of the sphere, using the relation [tex]\( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \)[/tex]:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{48}{2 \pi} \right) = \frac{4}{3} \cdot 24 = 32 \, \text{cm}^3
\][/tex]
So, the volume of the sphere is:
[tex]\[
\boxed{32 \, \text{cm}^3}
\][/tex]
