In this paper, we consider a UAV-assisted communication system comprising of single UAV serving to heterogeneous users having different data rate and coverage demands. Specifically, we propose a novel utility-aware transmission protocol to maximize the UAV utility by allowing it to simultaneously serve the highest possible number of users with available energy resources. In this regard, first we derive a closed-form expression for rate-coverage probability of a user considering Rician fading to incorporate the strong line of sight (LoS) component in UAV communication. Next, we formulate an optimization problem P to maximize the UAV utility under energy resources and rate-coverage constraints. Since, P is non-convex and combinatorial in nature, to provide global optimal solution, an equivalent distributed problem is formulated and a joint optimization algorithm is proposed which provide closed-form solution for joint-optimal power and time allocation. With the help of numerical investigation, we validate our coverage analysis and discuss the design insights on the optimal solution. We observe that the proposed joint-optimal resource allocation scheme can yield a significant gain in the UAV utility by making it to serve 60% more users as compared to benchmark fixed allocation scheme.