To solve this problem, follow these steps:
1. Identify the volume formulas:
- For a cylinder, the volume [tex]\( V_{\text{cylinder}} \)[/tex] is given by:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
- For a sphere, the volume [tex]\( V_{\text{sphere}} \)[/tex] is given by:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\][/tex]
2. Given data:
- The volume of the cylinder is [tex]\( 48 \, \text{cm}^3 \)[/tex].
- The sphere and cylinder share the same radius [tex]\( r \)[/tex] and the height [tex]\( h \)[/tex] of the cylinder is equal to the diameter of the sphere, which is [tex]\( 2r \)[/tex].
3. Express the height of the cylinder:
- From the volume of the cylinder, we know:
[tex]\[
\pi r^2 h = 48
\][/tex]
- Substitute [tex]\( h = 2r \)[/tex] into the cylinder's volume formula:
[tex]\[
\pi r^2 (2r) = 48
\][/tex]
[tex]\[
2 \pi r^3 = 48
\][/tex]
4. Solve for [tex]\( r^3 \)[/tex]:
- Isolate [tex]\( r^3 \)[/tex]:
[tex]\[
r^3 = \frac{48}{2 \pi}
\][/tex]
[tex]\[
r^3 = \frac{48}{2 \pi} = \frac{24}{\pi}
\][/tex]
[tex]\[
r^3 \approx 7.639437268410976
\][/tex]
5. Solve for [tex]\( r \)[/tex]:
[tex]\[
r = (r^3)^{1/3}
\][/tex]
[tex]\[
r \approx (7.639437268410976)^{1/3} \approx 1.969490043685393 \, \text{cm}
\][/tex]
6. Calculate the volume of the sphere:
- Using [tex]\( r \approx 1.969490043685393 \, \text{cm} \)[/tex]:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\][/tex]
- Substitute [tex]\( r^3 \approx 7.639437268410976 \)[/tex]:
[tex]\[
V_{\text{sphere}} = \frac{4}{3} \pi \times 7.639437268410976
\][/tex]
[tex]\[
V_{\text{sphere}} \approx 31.999999999999993 \, \text{cm}^3
\][/tex]
So, the volume of the sphere is [tex]\( 32 \, \text{cm}^3 \)[/tex], bearing in mind that we approximated the result to the nearest integer.
