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In rare-event detection wireless sensor network (WSN) applications maximizing lifetime and minimizing end-to-end delay (e-delay) are important factors in designing a cost-efficient network, which can be achieved using asynchronous sleep/wake (s/w) scheduling techniques and anycasting data-forwarding strategies, respectively. In this paper, we address the problem of finding an optimal cost WSN that satisfies given delay constraint and lifetime requirement, assuming random uniform deployment of nodes in a circular-shaped Field-of-Interest (FoI) and estimate the maximum of minimum expected e-delay in an anycasting forwarding technique for a given sensor density. We use this analysis to find the critical expected sensor density that satisfies given e-delay constraint, and lifetime requirement. We also validate our analysis.