Upon the application by petition of 50 or more taxpayers and residents of any union high school district to the board of supervisors in the county in which the union high school district 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.