Whenever a board of supervisors, by a four-fifths vote of its membership, determines that the public convenience and necessity demand the acquisition or construction of a new county highway or the improvement, repair or maintenance of any existing county highway, and that the expense of such new highway or the expense of improving, repairing or maintaining such existing highway is too great to pay out of the road fund of the district, the board may, by a resolution passed by a four-fifths vote of its membership, determine to make such acquisition or do such work, and charge the expense thereof to the county general fund, the road fund of the county or the district fund of any district benefited.