In this paper we address a sensor network lifetime optimization problem, where the network is given a task of reconstructing a Gaussian source with a specified rate-distortion constraint. The multiple agents observing the source relay their information through a multihop network to a sink or a base station. Although this problem has been addressed by other authors recently, only upper and lower bounds were obtained to the optimal lifetime as the nonlinear optimization problem was approximated by linearized constraints. We show that by a clever variable substitution, the original nonlinear optimization problem can be reformulated as a convex optimization problem and can be solved by sophisticated convex optimization tools. We provide numerical results comparing our optimal lifetime with the upper and lower bounds obtained in previous work and demonstrate that those bounds are not very tight. Some possible future research directions are mentioned in the concluding remarks.