Oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 5 million barrels of oil in the well; six years later 2,500,000 barrels remain.

a) At what rate was the amount of oil in the well decreasing when there were 3,000,000 barrels remaining?

b) When will there be 250,000 barrels remaining?

Using the decay law formula N=No x e^-kt

at t=0, No = 5 million barrels

at t = 6years, N = 2,500,000 barrels=No/2

a) the rate is 5 millions - 3,000,000=2,000,000 barrels

b) let t this time

so N=No x e^-kt, implies 250,000=5,000,000, x e^-kt

25=500, x e^-kt, 5=100, x e^-kt, we must search for k,

at t=0, No = 5 million barrels

at t = 6years, N = 2,500,000 barrels=No/2

so N=No/2=No x e^-kt implies 1/2=e^-6k, so ln (1/2) = lne^ - 6k, implies so

-0.69 = - 6k and k = 0.11

finally we can use N=No x e^-0.11t

250,000 = 5,000,000 x e^-0.11t

1 = 20 x e^-0.11t and 1/20 = e^-011t, 0.05 = e^-0.11t, ln (0.05) = - 0.11t

-3 = - 0.11t, so t = 27.22 years