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 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=

Explanation / Answer

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

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