This paper studies the problem of simultaneously locating trauma centers and helicopters. The standard approach to locating helicopters involves the use of helicopter busy fractions to model the random availability of helicopters. However, busy fractions cannot be estimated a priori in our problem because the demand for each helicopter cannot be determined until the trauma center locations are selected. To overcome this challenge, we endogenize the computation of busy fractions within an optimization problem. The resulting formulation has nonconvex bilinear terms in the objective, for which we develop an integrated method that iteratively solves a sequence of problem relaxations and restrictions. Specifically, we devise a specialized algorithm, called the shifting quadratic envelopes algorithm, that (1) generates tighter outer approximations than linear McCormick envelopes and (2) outperforms a Benders-like cut generation scheme. We apply our integrated method to the design of a nationwide trauma care system in Korea. By running a trace-based simulation on a full year of patient data, we find that the solutions generated by our model outperform several benchmark heuristics by up to 20%, as measured by an industry-standard metric: the proportion of patients successfully transported to a care facility within one hour. Our results have helped the Korean government to plan its nationwide trauma care system. More generally, our method can be applied to a class of optimization problems that aim to find the locations of both fixed and mobile servers when service needs to be carried out within a certain time threshold.