At the time and place prescribed in the said notice any owner of land within said proposed improvement district may file with the board his or her written consent to the proposed improvement, and he or she shall then be considered as a petitioner; and if the owners of more than one half of the lands within the district, including the lands represented by the petition, shall assent to the prayer of said petition, the board shall then proceed to hear and consider any objections which may have been filed at that or any previous time, and may adjourn such hearing from day to day. If the board after full hearing on the merits of the proposed waterway shall be satisfied that the same will be of benefit to the public interests, and that private benefit will result to the lands within the district sufficient to equal the cost of the proposed improvement, they may make findings accordingly and declare their intention to establish the waterway district under the name of the ". . . . . . Waterway District" and make the improvement as prayed for; but if the owners of less than one half of the lands in the district shall assent to the creation thereof and the making of the proposed improvement, the board shall deny the petition and the proceeding shall be dismissed.