To solve the problem, we need to determine the volume of a cylinder that a given cone fits perfectly inside of. Here are the steps for the solution:
1. Understand the Volume of the Cone:
- The given volume of the cone is 6 cubic inches.
2. Understand the Geometric Relationship:
- When a cone fits exactly inside a cylinder, they share the same base radius and height.
- The volume [tex]\( V \)[/tex] of a cone with radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] is given by:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\][/tex]
- The volume [tex]\( V \)[/tex] of a cylinder with the same radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] is:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
3. Relate the Cone's Volume to the Cylinder's Volume:
- Since both the cone and the cylinder have the same base radius and height, the volume of the cylinder is exactly three times the volume of the cone.
- Mathematically:
[tex]\[
V_{\text{cylinder}} = 3 \times V_{\text{cone}}
\][/tex]
4. Calculate the Cylinder's Volume:
- Given that the volume of the cone is 6 cubic inches, the volume of the cylinder will be:
[tex]\[
V_{\text{cylinder}} = 3 \times 6 = 18 \text{ cubic inches}
\][/tex]
Therefore, the volume of the cylinder that the cone fits exactly inside of is 18 cubic inches.
