Dropping that UV sphere on the base was pretty fun.
I first made the sphere on the mesh object level to set its origin to the bottom. Then, to align the sphere and the base, with 3D cursor snapped to top plane of the base mesh, I snapped the UV sphere to the 3D cursor. With two parts aligned, I merge two meshes into one object. Finally, I loop selected the bottom ring of the sphere and scaled on horizontal plane using vertex snapping with the width of the base. Merged by distance and two parts are thus joined.
