To find the volume of the sphere given that the sphere and cylinder have the same radius and height, and that the volume of the cylinder is [tex]\( 27 \pi \, \text{ft}^3 \)[/tex], let's solve the problem step-by-step.
1. Understanding the Cylinder's Volume:
The volume of a cylinder is given by the formula:
[tex]\[
V_\text{cylinder} = \pi r^2 h
\][/tex]
We know from the problem that:
[tex]\[
V_\text{cylinder} = 27 \pi \, \text{ft}^3
\][/tex]
2. Relating the Radius and Height:
The problem states that the sphere and cylinder have the same radius and height. For a sphere, the height in the context gets interpreted as the diameter of the sphere, which is 2 times the radius ([tex]\( h = 2r \)[/tex]).
3. Equating the Volumes:
Since the volume of the cylinder is [tex]\( \pi r^2 h \)[/tex] and we are given that [tex]\( h = 2r \)[/tex], we can substitute this into the cylinder volume formula:
[tex]\[
\pi r^2 (2r) = \pi r^2 \cdot 2r = 2\pi r^3
\][/tex]
Given that this volume equals [tex]\( 27 \pi \)[/tex]:
[tex]\[
2\pi r^3 = 27 \pi
\][/tex]
Dividing both sides by [tex]\( \pi \)[/tex] to simplify:
[tex]\[
2r^3 = 27
\][/tex]
4. Solving for the Radius:
Next, solve for [tex]\( r^3 \)[/tex]:
[tex]\[
r^3 = \frac{27}{2} = 13.5
\][/tex]
Therefore, [tex]\( r \)[/tex] can be found as:
[tex]\[
r = \sqrt[3]{13.5}
\][/tex]
5. Calculating the Volume of the Sphere:
The volume of a sphere is given by the formula:
[tex]\[
V_\text{sphere} = \frac{4}{3} \pi r^3
\][/tex]
Substituting [tex]\( r^3 = 13.5 \)[/tex]:
[tex]\[
V_\text{sphere} = \frac{4}{3} \pi (13.5)
\][/tex]
6. Simplifying the Sphere's Volume:
Combine the constants:
[tex]\[
V_\text{sphere} = \frac{4}{3} \pi \times 13.5 = \frac{4 \times 13.5}{3} \pi = \frac{54}{3} \pi = 18 \pi \, \text{ft}^3
\][/tex]
This value (18π) does not seem consistent with one of the given choices directly; let's use another interpretation quickly in context:
Checking \(V = \frac{2}{3} (27 \pi):
[tex]\[
V = \frac{2}{3} \times 27 \pi = 18 \pi \, \text{ft}^3
\][/tex]
Therefore, the volume of the sphere using given valid context and matching is :
[tex]\[
V = \frac{2}{3}(27 \pi)
\][/tex]
So, the correct equation for the volume of the sphere is:
[tex]\[
V = \frac{2}{3}(27 \pi)
\][/tex]
