Upon the application, by petition, of 50 or more taxpayers and residents of any unincorporated town or village to the board of supervisors in the county in which the town or village is located, for the formation of a library district, and setting forth the boundaries of the proposed district, the board of supervisors shall, within 10 days after receiving the petition, by resolution, order that an election be held in the proposed district for the determination of the question and shall conduct the election.