Question

Oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 3 million barrels of oil in the well; six years later 1,500,000 barrels remain.

(a) At what rate was the amount of oil in the well decreasing when there were 1,800,000 barrels remaining?

Answer 1

V' = -0.11552 *V\\= -0.11552(1.8)\\ \\=-0.20794 million per year

Explanation:

Given that oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 3 million barrels of oil in the well; six years later 1,500,000 barrels remain.

i.e. if V stands for volume of oil, then

`V' =kV\\dv/V= kdt\\ln V = kt+C\\V = Ae^(kt)`

To find A and k

V(0) = A = 3 million

Hence V =
`3e^(kt)`

V(6) = 1.5

i.e.
`1.5 = 3e^(6k)\\ln 1.5 =ln 3 +6k\\k = -0.11552`

`V(t) = 3e^(-0.11552t)`

a) Using the above value of k , we have

`V' = -0.11552 *V\\= -0.11552(1.8)\\ \\=-0.20794` million per year.