Patients shall be admitted to such hospitals in accordance with policies to be proposed by the board of trustees and approved by the county legislative authority. The policies shall provide, within the resources available to the hospital, that admission of patients shall not be dependent upon their ability to pay. Whenever a patient has been admitted to the hospital and in accordance with rules established by the board of trustees, the hospital may determine the person's ability to pay for the care provided by the hospital, render billings for the care, and take necessary steps to obtain payment for the costs of the care from the person, from the person's estate, or from any persons or organizations legally liable for the person's support.