  In this paper we consider complete noncompact Riemannian manifolds $(M, g)$
with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n
\geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces
$\partial \Omega$ in $M$, with equality holding true if and only if
$(M{\setminus}\Omega, g)$ is isometric to a truncated cone over
$\partial\Omega$. An optimal version of Huisken's Isoperimetric Inequality for
$3$-manifolds is obtained using this result. Finally, exploiting a natural
extension of our techniques to the case of parabolic manifolds, we also deduce
an enhanced version of Kasue's non existence result for closed minimal
hypersurfaces in manifolds with nonnegative Ricci curvature.
