Whenever a county commission shall desire that a state road or bridge on a state road in said county be constructed or maintained with state aid, written application shall be made by the county to the State Department of Transportation under such rules and regulations as the department may prescribe. Such application, when made, shall be considered by the department and, if approved by it, the commissioners shall direct an engineer to view said road or bridge and cause to be made surveys, plans, specifications, and estimates of the cost of construction or maintenance, and the State Department of Transportation may thereupon appropriate out of the State Highway Fund such part of the estimated cost of such work as it may deem proper, and the State Department of Transportation shall proceed to do such work by contract or with its own force. If it deems best, the department may accept appropriations from the county for said work, which shall be paid into the State Treasury to the credit of the State Highway Fund, before the work begins. Whenever a county fails to make application for the construction or maintenance of a road or bridge or the department deems it best for such work to be done, it may proceed to construct or maintain any part of the state road or bridge upon a state road and pay part or all of the cost of such work out of the State Highway Fund.