If we assume that the world population grows exponentially, then it is also reasonable to assume that the use of some nonrenewable resource (such as petroleum) will also grow exponentially. In calculus it is shown that for some constant \(k\), under these assumptions, the formula for the amount of the resource, \(A\), consumed from time \(t=0\) to \(t=T\) is given by the formula

\[A=\frac{A_{0}}{k}\left(e^{r T}-1ight)\]

where \(r\) is the relative growth rate of annual consumption.

a. Solve this equation for \(T\) to find a formula for life expectancy of a particular resource.

b. According to the Energy Information Administration, the annual world production (in billions of barrels per day) of petroleum is shown in the following table:

Find an exponential equation for these data.

c. If in 1998 the world petroleum reserves were 2.8 trillion barrels, estimate the life expectancy for petroleum. about 15 years

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Year 1975 1980 Quantity 52.42 62.39 1985 52.97 1990 1995 2000 2003 60.90 61.85 66.03 67.00