Health-care spending despite efforts at cost containment

Question

Question: HEALTH-CARE SPENDING Despite efforts at cost containment, the cost of the Medicare program is increasing. Two major reasons for this increase are an aging population and extensive use by physicians of new technologies. Based on data from the Health Care Financing Administration and the U.S. Census Bureau, health-care spending through the year 2000 may be approximated by the function

s(t) = 0.02836t³ - 0.05167t² + 9.60881t + 41.9         (0 ≤ t ≤ 35)

Where S(t) is the spending in billions of dollars and t is measured in years, with t 0 corresponding to the beginning of 1965.

a. Find an expression for the rate of change of health-care spending at any time t.

b. How fast was health-care spending changing at the beginning of 1980? At the beginning of 2000 ?

c. What was the amount of health-care spending at the beginning of 1980? At the beginning of 2000 ?