A sphere with a 3-inch radius has a 6-inch diameter and can be inscribed in a cube with at least side length 6.  Thus the smallest possible cube has side length 6 and volume $6^3=\boxed{216}$ cubic inches.

[asy]
size(60);
draw(Circle((6,6),4.5));
draw((10.5,6)..(6,6.9)..(1.5,6),linetype("2 4"));
draw((10.5,6)..(6,5.1)..(1.5,6));
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((0,9)--(3,12)--(12,12)--(9,9));
draw((12,12)--(12,3)--(9,0));
draw((0,0)--(3,3)--(12,3),dashed); draw((3,3)--(3,12),dashed);
[/asy]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
A steel sphere with a 3-inch radius is made by removing metal from the corners of a cube that has the shortest possible side lengths. How many cubic inches are in the volume of the cube?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
216
A steel sphere with a 3-inch radius is made by removing metal from the corners of a cube that has the shortest possible side lengths. How many cubic inches are in the volume of the cube?
test