We use numerical methods to obtain moduli-dependent Calabi-Yau metrics, and from them, the moduli-dependent massive tower of Kaluza-Klein states for the one-parameter family of quintic Calabi-Yau manifolds. We then compute geodesic distances in their Kähler and complex structure moduli space using exact expressions from mirror symmetry, approximate expressions, and numerical methods, and we compare the results. Finally, we fit the moduli dependence of the massive spectrum to the geodesic distance to obtain the rate at which states become exponentially light. The result is indeed of order 1, as suggested by the swampland distance conjecture. We also observe level crossing in the eigenvalue spectrum and find that states in small irreducible representations of the symmetry group tend to become lighter than states in larger irreducible representations.