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=(1)/(8)y_0`

and

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

we substitute the values

`(1)/(8)y_0=y_oe^(-0.09t)`
`(1)/(8)=e^(-0.09t)`

then we take the natural logarithmic of both sides

`\ln ((1)/(8))=\ln (e^(-0.09t))`

we simplify

`\ln ((1)/(8))=-0.09t`

then we isolate the t

`t=(\ln ((1)/(8)))/(-0.09)`

then we obtain the value of t

`t=23.1049`

to the nearest year t=23 years