§ 49. Improvement of state highways at expense of county or at the joint expense of such county and any city, village or town. The board of supervisors of any county may request of the department an estimate of the additional cost of improving a state highway to a width or in a manner, or both, as desired by such board. The department shall as soon as practicable make an estimate of such additional cost and transmit the same to the board of supervisors, and the board of supervisors may thereupon by resolution petition the commissioner of transportation to provide the width and type of improvement desired. The expense caused by the increased width or the type of improvement, or both, shall be borne wholly by the county, or, in the discretion of the board of supervisors of any county, and with the consent of the appropriate board of any city, village or town in which the road is located, the expenses may be apportioned between such county and such other municipality or municipalities, in such proportion as may be agreed to. Whenever plans and specifications have been approved providing for such increased width or such type of improvement, or both, an estimate of the cost of such increased width or type of improvement, or both, shall be submitted to the board of supervisors, which, if it approve such plans, specifications and estimate of cost, shall, by resolution, duly adopted by a vote of a majority of all of the members of such board, appropriate the funds necessary to provide therefor; and such funds shall be made available for the commissioner of transportation in the manner provided in this article.