The one-dimensional (slab), one-velocity time-dependent transport equation has been investigated using a variational method employing flat spatial trial functions. A simple approximation is found for the variation of the asymptotic decay rate (α) with slab size for small slabs. As expected, little difference is found between the use of a single flat spatial flux trial function and a double stepped flux trial function for thin slabs. The method is then extended to the case of a convex body of arbitrary shape. It is shown that an estimate for α is given by the relation where Pc = first collision probability. For the slab case, an effective spatial buckling and an effective extrapolation distance consistent with the exact asymptotic decay constant were obtained. This extrapolation distance is approximately equal to the Milne problem value down to a scattering thickness of about 1.0 mean free path after which it rises to λs for the limiting case of zero thickness. Finally, asymptotic time decay rates based upon low-order PL and DPL approximations in slab geometry are determined either numerically or from the exact analytical solutions; a real eigenvalue may or may not exist depending on the boundary conditions. It is shown further that these low-order approximations yield erroneous time-dependent characteristics in the thin slab limit.