Question

Suppose the amount of oil pumped from a well decreases at a continuous rate of 9% per year. When, to the nearestyear, will the well's output fall to 1/8 of its present value?

Answer 1

We will use the next formula

$y=y_0e^{-kt},_{}k>0_{}$

where

$y=\frac{1}{8}y_0$

and

$k=9\text{\%=0.09}$

we substitute the values

$\frac{1}{8}y_0=y_oe^{-0.09t}$$\frac{1}{8}=e^{-0.09t}$

then we take the natural logarithmic of both sides

$\ln (\frac{1}{8})=\ln (e^{-0.09t})$

we simplify

$\ln (\frac{1}{8})=-0.09t$

then we isolate the t

$t=\frac{\ln (\frac{1}{8})}{-0.09}$

then we obtain the value of t

$t=23.1049$

to the nearest year t=23 years