Question

answer the following, show work.

Suppose that the world population grows exponentially. Then it is reasonable to assume that the use of nonrenewable resources, such as petroleum and coal, also grows exponentially. Under these conditions, the following formula can be derived: where A is the amount of resource consumed from time t=0 to t=T, A0 is the amount consumed during the year t0 and k is the relative (annual) growth rate (of consumption). The inverse function T = f(-1)(A) is called the life expectancy for a given resource. Solve for T. Data from the years 1965 to 1972 indicate that world oil consumption followed this exponential model. Let t = 0 correspond to 1972. In that year worldwide consumption of oil was approximately 18.7 billion barrels, with a relative growth rate of 7 percent per year (k = .07). The global reserves at that time were estimated to be 700 billion barrels. Use this data to compute the life expectancy T of oil under this model and specify the depletion year (1972+T) T, the depletion year is a function of k, the relative growth rate. Find dT/dk. As k decreases, what happens to the value of dT/dk? If the consumption growth rate is reduced to 3.5 percent per year, compute T.