Oil is pumped continuously from a well at a rate proportional to the amount of o

Question

Oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 4 million barrels of oil in the well; six years later 2,000,000 barrels remain.

(a) At what rate was the amount of oil in the well decreasing when there were 2,400,000 barrels remaining?
rate = ? barrels/year

(b) When will there be 200,000 barrels remaining?
after= ? years.

Explanation / Answer

Here let the quantity of oil any any time be O. Then dO/dt = -kO => dO /O = -kdt => ln O = -kt +C at t=0, O = 4 million , so C = ln 4 million so ln (O/Oo) = -kt Oo = 4 million , also given , at t= 6 , O= 2 million, so ln (0.5)= -k*6 => k = 0.115 a) when O = 2.4 million , dO/dt = -k*O =0.115*2.4 = 0.277million barrels/ year b) ln ( 0.2/4) = -0.115*t t=25.93 years

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