The board of trustees shall:

(1) Have general supervision and care of such hospitals and institutions and the buildings and grounds thereof and power to do everything necessary to the proper maintenance and operation thereof within the limits of approved budgets and the appropriations authorized;

(2) Elect from among its members a president and vice president;

(3) Adopt bylaws and rules for its own guidance and for the government of the hospital;

(4) Prepare annually a budget covering both hospital operations and capital projects, in accordance with the provisions of applicable law, and file such budgets with the county treasurer or if the hospital has been established by more than one county, with the county treasurer of each county, and if a city has contributed to the establishment of the hospital, with the official of the city charged by law with the preparation of the city budget; and

(5) File with the legislative authority of each county and city contributing to the establishment of such hospital, at a time to be determined by the county legislative authority of the county in which the hospital is located, a report covering the proceedings of the board with reference to the hospital during the preceding twelve months and an annual financial report and statement.