Whenever the owners of not less than one-half of the land bordering on or adjacent to any section of road established shall petition for a survey and estimate of the cost of building the road in a substantial and permanent manner of stone, gravel, chert, slag, brick, a combination of such material, or of any material equally durable and economical, the board may cause a survey and estimate to be made by a licensed civil engineer. Whenever thereafter the petitioners shall present to the board a map or description of the land which in their opinion would be directly benefited by the construction of such road, together with the written requests of the owners of not less than three-fifths of such land asking that all lands so benefited be assessed in proportion to the benefit conferred to pay one-third of the cost of construction, the board may levy and collect a special tax on such lands sufficient to pay one third of the cost thereof. Such lands shall be known as the benefit district of such section of road.