In the event no appeal is perfected as provided in section 31-911, or upon final determination of said appeal, unless the court should hold that the improvement is not conducive to public health, convenience and welfare, the board shall proceed as follows:
(1) If the estimated cost of the improvement is two thousand dollars, or less, the board may, through rental of equipment or private contract, make said improvements without public letting;
(2) If the estimated cost of the improvement is more than two thousand dollars, plans and specifications therefor shall be drawn by an engineer employed by said board. At such time as the plans and specifications are complete, the board shall fix a time for the opening of bids on said improvement and publish notice thereof for three weeks prior to such letting;
(3) No project shall be divided into small sections in order to avoid the requirement for public letting set forth in this section; and
(4) No bid shall be accepted which is more than the estimated cost of said improvement. The board shall let the contract to the lowest responsible bidder and shall take from said bidder a sufficient bond as security for performance of the contract pursuant to and within the time fixed by said plans and specifications.