Question: We analyze world oil production. When annual world oil production peaks and starts to decline, major economic restructuring will be needed. We investigate when this slowdown is projected to occur.
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.