Whenever a petition shall be presented to the board of county commissioners of any county of this state praying for a public highway, and the names of all the owners of all the land through which said road is to be laid out, shall be signed by the owners thereof to said petition, giving the right-of-way through the land, and accompanied by a plat of the road, it shall be the duty of the board of county commissioners, if in their opinion, the public good requires it, to declare the same a public highway, and thereupon a plat shall be filed and recorded and the said road shall become a public highway from and after that date.