Answer:
V' = -0.11552 *V\\= -0.11552(1.8)\\ \\=-0.20794 million per year
Step-by-step 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
[tex]V' =kV\\dv/V= kdt\\ln V = kt+C\\V = Ae^{kt}[/tex]
To find A and k
V(0) = A = 3 million
Hence V = [tex]3e^{kt}[/tex]
V(6) = 1.5
i.e. [tex]1.5 = 3e^{6k}\\ln 1.5 =ln 3 +6k\\k = -0.11552[/tex]
[tex]V(t) = 3e^{-0.11552t}[/tex]
a) Using the above value of k , we have
[tex]V' = -0.11552 *V\\= -0.11552(1.8)\\ \\=-0.20794[/tex] million per year.
