Whenever one-third of the resident taxpayers of any county, as appears by the last preceding assessment roll of the county, petition the board of county commissioners requesting an appropriation to build a bridge across any navigable river on the line of the county, setting forth therein the location of the bridge as near as may be, its estimated cost, the necessity therefor to accommodate the general traveling public, the manner in which it is proposed to pay for the structure, and the time when it will be completed, the petition to be duly verified by the affidavits of at least fifteen of the petitioners therein named, the board of county commissioners shall publish a notice in the official newspaper of the county, once each week for three consecutive weeks, briefly stating the object of the petition and that the same will be heard and considered at the next regular meeting of such board. At the time appointed for the hearing of the petition, the board of county commissioners shall investigate the need for such bridge, and if it finds the same to be necessary and that the consent of the federal government has been obtained to span such river, it, by resolution duly entered upon the minutes of the board, shall appropriate toward the building of such bridge, from the county treasury, a sum not exceeding one-half of the estimated cost of the bridge. The appropriation must be upon condition that a sufficient bond be given, conditioned that the remaining one-half or more, as the case may be, of the cost of the bridge will be paid.