  Given a convex cone in the \emph{prescribed} warped product, we consider
hypersurfaces with boundary which are star-shaped with respect to the center of
the cone and which meet the cone perpendicularly. If those hypersurfaces inside
the cone evolve along the inverse mean curvature flow, then, by using the
convexity of the cone, we can prove that this evolution exists for all the time
and the evolving hypersurfaces converge smoothly to a piece of round sphere as
time tends to infinity.
