When a large amount of sensors are randomly deployed into a field, how can we make a sleep/activate schedule for sensors to maximize the lifetime of target coverage in the field? This is a well-known problem, called Maximum Lifetime Coverage Problem (MLCP), which has been studied extensively in the literature. It is a long-standing open problem whether MLCP has a polynomial-time constant-approximation. The best-known approximation algorithm has performance ratio 1 + ln n where n is the number of sensors in the network, which was given by Berman et. al . In their work, MLCP is reduced to Minimum Weight Sensor Coverage Problem (MWSCP) which is to find the minimum total weight of sensors to cover a given area or a given set of targets with a given set of weighted sensors. In this paper, we present a polynomial-time (4 + ε)-approximation algorithm for MWSCP and hence we obtain a polynomial-time (4+ξ)-approximation algorithm for MLCP, where ε > 0, ξ > 0.