The council or governing body of the city shall have the same rights and powers by and under an ordinance enacted to assess and collect a street tax from every person liable to road or street duty residing in the territory exempt from taxation that it has to assess and collect such tax from other persons residing within the corporate limits of the city, but not in territory exempt from taxation.

The money collected from persons residing in territory exempt from taxation shall be applied by the council or governing body of the city towards keeping up and maintaining the public roads or streets in the territory exempt from taxation or in opening new roads or streets in such territory.