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=

Answer 1

Explanation:

From the information we find that

Q'(t) = kQ(t)

with Q(0) = 5 and Q(6) = 2.5 in millions

A) Separate the variables and solve the DE

`(dQ)/(Q) =kdt\\ln Q = kt+C\\Q = Ae^(kt)`

Using Q(0) = 5

A =5

B)
`Q = 5e^(kt)`

C)

Using Q(6) = 2.5

`2.5 =5e^(6k) \\ln 0.5 = 5k\\k=-0.1155`

D) Use Q'(t) = -0.1155 (3) = -0.3465

Rate = -346500 barrels

(negative indicates decrease in number of barrels_

E) When Q = 250000 we have

`0.25 = 5e^(-0.1155t) \\-0.1155 t = -2.996\\t = 25.93`

After 26 years