A tournament \(T = (V, A)\) is \({arc-traceable}\) if for each arc \(xy \in A\), \(xy\) lies on a directed path containing all the vertices of \(V\), i.e., a hamiltonian path. In this paper, we give two extremal results related to arc-traceability in tournaments. First, we show that a non-arc-traceable tournament \(T\) which is \(m\)-arc-strong must have at least \(2^{m+1}+4m-3\) vertices, and we construct an example that shows that this result is best possible. Next, we consider the maximum number of arcs in a strong tournament that are not part of any hamiltonian path. We use the structure of non-arc-traceable tournaments to prove that no strong tournament contains more than \(\frac{n^2-4n+3}{8}\) arcs that are not part of a hamiltonian path, and we give the unique example that shows that this bound is best possible.