This paper focuses on the application using a solar-powered unmanned aerial vehicle (UAV) to inspect mountain sites for the purpose of safety and rescue. An inspection path planning problem is formulated, which looks for the path for an UAV to visit a set of sites where people may appear while avoiding collisions with mountains and maintaining positive residual energy. A rapidly exploring random tree (RRT)-based planning method is proposed. This method firstly finds a feasible path that satisfies the residual energy requirement and then shortens the path if there is some abundant residual energy at the end. Computer simulations are conducted to demonstrate the performance of the proposed method.