We construct a mathematical model of the within-cell replication of poliovirus, a prototypic RNA virus, and use realistic parameter estimates to describe the increase of copy number of the viral genome. Our initial model is essentially an exponential growth model; we also consider modifications of this model to account for resource utilization. The saturation of viral replication dynamics observed in experimental systems can be explained in terms of heavy resource use by the virus. We then use our models to consider the conditions under which the growth of poliovirus is optimized. Intriguingly, if poliovirus has optimized its replication within cells, the predicted ratio of positive to negative strands is close to what is actually observed. We interpret our findings in terms of the evolution of life-history traits.