The board of county commissioners, upon receipt of a petition signed by at least ten percent of the qualified electors of the county as determined by the number of votes cast for the office of governor at the preceding general election, shall, in accordance with applicable provisions of title 16.1, cause the question of whether commissioners shall be elected at large to be submitted to the qualified electors of the county at the next succeeding primary or general election. If approved by sixty percent of the qualified electors voting at such election, all county commissioner districts in the county must be immediately dissolved, and thereafter as the term of office of each member of the board of county commissioners expires, the office must be filled by an election at large.