(A) It shall be the duty of the City Engineer to cause to be kept a just and true account of all expenses incurred by his or her department in connection with such improvements.
(B) It shall be the duty of the City Engineer to cause such improvements made by or under private contract to be duly inspected and supervised during the progress of the work thereupon and to cause to be kept a just and true account of the cost of inspection and supervision of such improvements, and otherwise to carry out the provisions of this chapter.
(`83 Code, § 12.04.030)