Homework SourseOil is pumped continuously from a well at a rate proportiona
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):Let Q(t) be the number of barrels left in the well after t years, measured in millions. Write a differential equation for Q that captures the information in the problem. Use k>0 for any proportionality constant you need. Be careful about signs.
Q\'=
(B):Solve the above differential equation, without yet determining k.
Q(t)=
(C):Determine k.
k=
(D):At what rate was the amount of oil in the well decreasing when there were 3,000,000 barrels remaining? [Hint: Use the equation in (a). Be careful about units.]
rate= barrels/year
(E):When will there be 250,000 barrels remaining?
years=
(A):Let Q(t) be the number of barrels left in the well after t years, measured in millions. Write a differential equation for Q that captures the information in the problem. Use k>0 for any proportionality constant you need. Be careful about signs.
Q\'=
(B):Solve the above differential equation, without yet determining k.
Q(t)=
(C):Determine k.
k=
(D):At what rate was the amount of oil in the well decreasing when there were 3,000,000 barrels remaining? [Hint: Use the equation in (a). Be careful about units.]
rate= barrels/year
(E):When will there be 250,000 barrels remaining?
years=
Solution
A)dQ/dt =KQ
SOLUTIONFOR THIS DIFFE EQUATION IS
Q(t)=Q(0)ekt
b)Q(0)=5 millon barrels ,t=6 years
Q(t)=2,500,000 barrels =2.5millon barrels
2.5=5e6k
6k=ln(1/2)=-0.69
K=-0.1155
therfore Q(t)=5e-0.1155t
c)K=-0.1155
d)dQ/dt = kQ
=-0.1155*3=-0.3465
e)Q(t)=250,000=0.25
0.25 = 5e-0.1155t
t=25.93 years
