  We prove that there exists a gradient expanding Ricci soliton asymptotic to
any given cone over the product of a round sphere and a Ricci flat manifold. In
particular we obtain asymptotically conical expanding Ricci solitons with
positive scalar curvature on $\mathbb{R}^3 \times S^1.$ More generally we
construct continuous families of gradient expanding Ricci solitons on trivial
vector bundles over products of Einstein manifolds with arbitrary Einstein
constants.
