Any petition for the formation of a public hospital capital facility area may describe an area less than the entire county in which the petition is filed, the boundaries of which must follow the then existing precinct boundaries and not divide any voting precinct; and in the event that a petition is filed containing not less than ten percent of the voters of the proposed public hospital capital facility area who voted at the last general election, certified by the auditor in like manner as for a countywide district, the board of county commissioners shall fix a date for a hearing on the petition, and shall publish the petition, without the signatures thereto appended, for two weeks prior to the date of the hearing, together with a notice stating the time of the meeting when the petition will be heard. Publications required by this chapter must be in a newspaper published in the proposed public hospital capital facility area, or, if there be no such newspaper, then in a newspaper published in the county in which the public hospital capital facility area is situated, and of general circulation in that county. The hearing on the petition may be adjourned from time to time, not exceeding four weeks in all. If upon the final hearing the board of county commissioners finds that any lands have been unjustly or improperly included within the proposed public hospital capital facility area the board shall change and fix the boundary lines in such manner as it deems reasonable and just and conducive to the welfare and convenience, and make and enter an order establishing and defining the boundary lines of the proposed public hospital capital facility area: PROVIDED, That no lands may be included within the boundaries so fixed lying outside the boundaries described in the petition, except upon the written request of the owners of those lands.