Okay, let's try to figure out this problem. So, we have a steel sphere with a 3-inch radius, and it's made by removing metal from the corners of a cube with the shortest possible side lengths. We need to find the volume of the cube. Hmm, okay. Let me think step by step.

First, the sphere is made by removing metal from the corners of the cube. That means the sphere is inscribed within the cube, right? Because if you take a cube and remove metal from each corner, the largest sphere that can fit inside the cube would touch the centers of each face of the cube. So, the sphere is inscribed in the cube. Therefore, the diameter of the sphere should be equal to the side length of the cube. Wait, but the radius is given as 3 inches. So, the diameter would be twice that, which is 6 inches. That would mean the cube has a side length of 6 inches. Then, the volume of the cube would be side length cubed, so 6^3 = 216 cubic inches. But wait, that seems too straightforward. Let me check if that's correct.

Wait, if the sphere is inscribed in the cube, then the diameter of the sphere is equal to the edge length of the cube. So, if the radius is 3 inches, diameter is 6 inches, so cube edge is 6 inches. Volume is 6*6*6=216. But the problem says "the shortest possible side lengths." So perhaps there's a different consideration here. Maybe the sphere isn't inscribed in the cube? Because if you remove metal from the corners, the sphere might not be touching all the faces of the cube. Hmm, maybe the sphere is inscribed in the cube in a different way, or maybe the cube is circumscribed around the sphere. Wait, no, if you remove metal from the corners, the sphere would fit perfectly inside the cube, touching the centers of each face. So, the sphere is inscribed in the cube. Therefore, the diameter of the sphere is equal to the edge length of the cube. So, radius 3, diameter 6, edge length 6, volume 216. But that seems too simple, and the problem mentions "shortest possible side lengths," which might imply that the cube isn't just any cube but the minimal one that can contain the sphere. But if the sphere is inscribed, then the cube's edge is fixed by the sphere's diameter. So perhaps that's the answer. Wait, but maybe the sphere is not inscribed but is something else. Let me think again.

Alternatively, perhaps the cube is such that the sphere is formed by truncating the cube's corners. When you truncate a cube's corners, you create a sphere-like shape, but in reality, it's a truncated cube, which has flat faces where the corners were removed. However, the problem says the sphere is made by removing metal from the corners. So, maybe the resulting shape is a sphere, but how? If you remove metal from each corner of the cube, you end up with a shape that has flat faces where the corners were removed, but the original faces of the cube are still there. Unless the removal is done in such a way that the resulting shape is a sphere. That seems impossible because a cube has flat faces and right angles, and a sphere is perfectly round. The only way to make a sphere from a cube is if the cube is somehow transformed into a sphere, but removing metal from the corners would leave the original faces intact, which are flat. So maybe the problem is saying that the sphere is the remaining part after removing metal from the cube's corners. Wait, but then the sphere would be inside the original cube, but how? If you remove metal from the corners, the remaining part would have indentations at the corners, but the overall shape would still have the original cube's faces, which are flat. So unless the sphere is the convex hull of the remaining parts, but that's just the original cube. Hmm, I'm confused.

Wait, maybe the problem is that the sphere is formed by cutting off the corners of the cube, turning each corner into a small sphere segment. But that doesn't make sense because cutting off a corner of a cube creates a triangular face, not a spherical one. So perhaps the problem is misworded, or I'm misunderstanding it. Alternatively, maybe the sphere is the largest sphere that can fit inside the cube after truncating the corners. But then, the sphere would be tangent to the centers of the original cube's faces, which would again imply that the cube's edge is equal to the sphere's diameter. Therefore, edge length 6, volume 216. But the problem says "the shortest possible side lengths," which might mean that the cube is the minimal cube that can contain the sphere. But if the sphere is inscribed, then the cube's edge is fixed at 6. So maybe that's the answer. But let me check again.

Wait, perhaps the sphere is not inscribed but is such that the cube's edges are longer, and the sphere is contained within the cube, but the cube's corners are cut in such a way that the sphere touches the cube's edges. Wait, if you have a cube with edge length a, and you cut off each corner such that the resulting shape is a sphere, that doesn't seem possible. Alternatively, maybe the sphere is inscribed in the cube, and the cube's side length is equal to the sphere's diameter. So, radius 3, diameter 6, cube edge 6, volume 216. But then why mention "shortest possible side lengths"? Maybe there's a different way to fit the cube around the sphere with shorter edges? That doesn't make sense because the sphere's diameter defines the minimal cube that can contain it. If the sphere is inscribed, the cube's edge is exactly the sphere's diameter. So perhaps that's the case. Therefore, the volume is 6^3 = 216. But let me check if there's another interpretation.

Alternatively, maybe the sphere is formed by truncating the cube's corners such that the resulting shape is a sphere, but that's not possible with simple truncation. Truncation would create flat faces, not spherical ones. Therefore, the only way to have a sphere is if the original cube is somehow transformed into a sphere by removing metal from the corners. But that's not possible unless you're doing something else. Therefore, perhaps the problem is that the sphere is inscribed in the cube, so the cube's edge is equal to the sphere's diameter, and the volume is 6^3=216. Therefore, the answer is 216. But let me check if that's correct.

Alternatively, maybe the sphere is the result of the cube's corners being removed such that the sphere touches the centers of the cube's edges. Wait, if the sphere touches the centers of the edges, then the radius would be different. Let me think: the distance from the center of the cube to the center of an edge is (a/2)*sqrt(2), where a is the edge length. So if the sphere touches those points, then the radius would be (a/2)*sqrt(2). Given that the radius is 3, then 3 = (a/2)*sqrt(2) => a = 6/sqrt(2) = 3*sqrt(2). Then the volume would be (3*sqrt(2))^3 = 27*(2*sqrt(2)) = 54*sqrt(2). But that's about 76.36, which is much smaller. But in that case, the sphere is touching the centers of the cube's edges, not the faces. But the problem says the sphere is made by removing metal from the corners, so it's more likely that the sphere is inscribed in the cube, touching the centers of the faces. Therefore, the edge length is 6, volume 216. But why mention "shortest possible side lengths"? Maybe the cube is the minimal cube that can contain the sphere after truncation, but if the truncation removes parts of the cube, the remaining part could have a smaller edge length. Wait, but truncating the cube's corners would create a shape that's a cube with its corners cut off, but the original cube's faces are still present. So the remaining part would still have the original cube's faces, which are flat, so the overall shape isn't a sphere. Therefore, perhaps the problem is that the sphere is inscribed in the cube, so the cube's edge is 6, volume 216. Therefore, that's the answer. But let me check again.

Alternatively, maybe the sphere is the part that remains after removing metal from the cube's corners, so the cube's corners are removed, and the sphere is the remaining part. But that doesn't make sense because the corners are removed, so the remaining part would have indentations, but the original cube's faces are still there. So the remaining part would still be a cube with smaller edges? No, the original cube's edges are intact, just the corners are removed. So the remaining part is still a convex polyhedron, not a sphere. Therefore, the only way the remaining part is a sphere is if the cube is transformed into a sphere by removing metal from the corners in such a way that the resulting shape is a sphere. But that's not possible with simple truncation. Therefore, maybe the problem is that the sphere is inscribed in the cube, so edge length 6, volume 216. Therefore, that's the answer. But the mention of "shortest possible side lengths" is confusing. Maybe the cube is the smallest cube that can contain the sphere after truncation. But if truncation removes parts, the remaining part is a cube with smaller edge length? But no, the original cube's edges are still there. Wait, maybe the cube is the minimal cube that can contain the sphere such that all the cube's corners are removed. But if the sphere is inside the cube, the cube's edge length must be at least the diameter of the sphere, which is 6. Therefore, the minimal cube is 6x6x6, so volume 216. Therefore, that's the answer. Therefore, the final answer is 216.

But wait, let me think again. If the cube is the minimal cube that can contain the sphere after truncation, then the cube's edge length is 6. Therefore, the volume is 6^3 = 216. Therefore, the answer is 216. So, I think that's the correct answer.
Therefore, the volume of the cube is $\boxed{216}$ cubic inches.
</think>