Chapter 7.EA, Problem 1EA

To determine

To find:

The number of years that the estimated petroleum reserves would last if used at the same rate as in the base year.

Answer to Problem 1EA

Solution:

The estimated petroleum reserves would last if used at the same rate as in the base year over $88$ years after $1970$.

Explanation of Solution

Approach:

1. Write equation for the rate of consumption of minerals at time $t$.
2. Use the property of integration as,

${\displaystyle \int kdt}=kt+k$ and $k$ is a real number.

Calculation:

The equation for the rate of consumption of minerals at time $t$ is,

$q=q_0{e}^{kt}$.

Here $q$ is the rate of consumption at time $t$, $q_0$ is the rate of consumption at time $t=0$, and $k$ is the growth rate constant.

The base year is $1970$ thus it corresponds to $t=0$.

The consumption of petroleum in the base tear is $16,900\text{million}\text{barrels}$.

The total consumption of petroleum is estimated as $1,500,000\text{million}\text{barrels}$.

The total consumption of mineral to $T$ years is,

$\text{Total Consumption}={\displaystyle {\int }_0^T{q}_0{e}^{kt}dt}$.

Substitute $q_0=16900$, $k=0$ and total consumption $1,500,000\text{million}\text{barrels}$.

$\begin{array}{rll}\text{Total Consumption}& =& {\displaystyle {\int }_0^{T}16900e^{k\left(0\right)}dt}\\ 1,500,000& =& {\displaystyle {\int }_0^{T}16900dt}\\ 1,500,000& =& {\left[16900t\right]}_0^T\\ 1,500,000& =& 16900T\\ T& =& 88.75\\ T& \approx & 89\text{years}\end{array}$

Therefore, the estimated petroleum reserves would last if used at the same rate as in the base year over $89$ years after $1970$.