Question

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? 

Answer 1

Let r be the rate at which oil production is decreasing from the well. Then:
5000000 x r^6=2500000
r^6=.5
ln r^6=ln .5
6 ln r=-0.69314718055994530941723212145818
ln r=-0.11552453009332421823620535357636
r=e^-0.11552453009332421823620535357636=.891
Since the oil production is only 0.891 of the previous year, the annual rate of loss is 0.109, or 10.9%, growing annually
A.3000000=5000000x(.891)^t
.6=0.891^t
t=4.42 years
Oil is being pumped at .60 of original production
B.250000=5000000 x 0.891^t
.05=.89^t
t=25.96 years
☺☺☺☺