This paper considers the potential for passive acoustic target detection using a maneuverable unmanned underwater vehicle (UUV) equipped with towed line array. The objective is to assess target detection performance as a function of vehicle maneuverability, environmental variability, and array length. Detection performance is evaluated by mapping the search region onto a graph wherein vehicles are constrained in their trajectories as function of their maximum turning angle. Detection range, as a function of location on the graph, is encoded on the edge weights between nodes. With this framework, the UUV trajectory which maximizes the probability of at least one target detection along a transit through the region can be determined using the computationally efficient Dijkstra algorithm. Increasing UUV maneuverability is shown to significantly improve detection performance given known environmental variability since the optimal trajectory is able to visit more favorable locations, or "hot spots" with respect to environmental conditions. Moreover, it is shown that in sufficiently variable, but known, environmental conditions, a maneuverable UUV with short array is competitive to a much longer non-maneuverable array.