The final surveys, plans, specifications, and detailed drawings shall include a final estimate of cost. If the final estimate exceeds by more than 10 per cent the estimate named in the preliminary report of the district engineer, the final surveys, plans, specifications, and detailed drawings shall not be approved by the board of directors until the increase in estimated costs is approved by the board of supervisors of each county comprising the district, and by the Director of Transportation if any contribution is to be made by the state.