Sec. 12. (a) This section applies to governing boards of a county hospital in a county having a population of more than:
(1) sixteen thousand seven hundred (16,700) and less than nineteen thousand (19,000);
(2) twenty-four thousand five hundred (24,500) and less than twenty-four thousand six hundred (24,600); or
(3) forty-six thousand four hundred (46,400) and less than forty-seven thousand (47,000).
(b) The appointing authority shall appoint a member to fill a vacancy on the governing board within sixty (60) days after the vacancy occurs.