Question: Use the logistic function obtained in Problem to model the growth of P, the total oil produced worldwide in billions of barrels since 1859:
(a) Find the projected value of P for 2010.
(b) Estimate the annual world oil production during 2010.
(c) How much oil is projected to remain in the ground in 2010?
(d) Compare the projected production in part (b) with the actual figure of 26.9 billion barrels.
Problem: We define P to be the total oil production worldwide since 1859 in billions of barrels. In 1993, annual world oil production was 22.0 billion barrels and the total production was P = 724 billion barrels. In 2008, annual production was 26.9 billion barrels and the total production was P = 1100 billion barrels. Let t be time in years since 1993.
(a) Estimate the rate of production, dP/dt, for 1993 and 2008.
(b) Estimate the relative growth rate, (1/P)(dP/dt), for 1993 and 2008.
(c) Find an equation for the relative growth rate, (1/P)(dP/dt), as a function of P, assuming that the function is linear.
(d) Assuming that P increases logistically and that all oil in the ground will ultimately be extracted, estimate the world oil reserves in 1859 to the nearest billion barrels.
(e) Write and solve the logistic differential equation modeling P.