My teacher gave me this problem, and it is very wordy, I don't really even understand what it is asking. First I took 100 and multiplied it by 0.13 subtracting that number from 100 and completing the process with the remaining left until that value was under 21. From this I got 12 year, but it was not right. Could anyone help me understand this problem?
A new oil field has just begun production. The first oil removed is the easiest to get out, and so production falls as time goes on. The instantaneous rate at which oil can be extracted is 13% of the amount of oil remaining per year. Here, "instantaneous" refers to the fact that as soon as any oil is removed, the rate of production falls proportionally in the "next" instant. If the company continues to extract oil at that instantaneous rate, when will the amount of oil left in the field first be less than 21 percent of the original amount? (in years)
If $y$ is the oil in the reservoir and $t$ is the time in years since extraction started, we have
$$\frac{dy}{dt} = -0.13y$$
Taking $R$ as the original reserve of oil, this is solved by
$$y = Re^{-0.13t}$$
Then you need to find $t$ such that
$$\begin{align}0.21R &= Re^{-0.13t} \\ 0.21 &=e^{-0.13t} \\ \end{align}$$
etc.