Chapter 7.4, Problem 82E

Oil Consumption Suppose that the rate of consumption of a natural resource is $c^{\prime }(t)$, where $c^{\prime }(t)=k{e}^{rt}$, Here $t$ is the time in years, $r$ is a constant, and $k$ is the consumption in the year when $t=0$. In $2010$, an oil company sold $1.2$ billion barrels of oil. Assume that $r=0.04$.

a. Write $c^{\prime }(t)$ for the oil company, letting $t=0$ represent $2010$.

b. Set up a definite integral for the amount of oil that the company will sell in the next $10$ years.

c. Evaluate the definite integral of part b.

d. The company has about $20$ billion barrels of oil in reserve. To find the number of years that this amount will last, solve the equation ${\displaystyle \underset{0}{\overset{T}{\int }}1.2e^{0.04t}dt}=20$.

e. Rework part d, assuming that $r=0.02$.