When will oil pumping no longer be profitable?

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In summary, the conversation discusses the continuous pumping of oil from a well at a rate proportional to the amount of oil remaining. An equation is given for the amount of oil remaining at any time t, y=y0e^(kt), and the initial and final amounts of oil are known. The rate at which the oil is decreasing and the optimal time to stop pumping in order to avoid losing money are also mentioned. The solution to finding the value of k is given, and it is noted that only the value of e^k is necessary for the equation.
  • #1
lreichardt
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1. Oil is being pumped continuously from a certain oil well at a rate proportional to the amount of oil left in the well; that is, dy/dt=ky, where y is the amount left in the well at any time t. Initially, there were 1,000,000 gallons of oil in the well, and 6 years later there were 500,000 gallons remaining. It will no longer be profitable to pump oil when there are fewer than 50,000 gallons remaining.

1.a Write an equation for y, the amount of oil remaining in the well at any time t.

Attempt at solution:

y=y0e^(kt)
I know I need to solve for k, I just don't know how.
At t=0, y0=1,000,000
t=6 (Half life=??)

1. b At what rate is the amount of oil in the well decreasing when there are 600,000 gallons of oil remaining?

solution: haven't quite gotten here yet...

1. c In order not to lose money, at what time t should oil no longer be pumped from the well?

solution: see above. :-(

Thank you to anyone who can point me in the right direction, here.
 
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  • #2
Right, so at t = 6,

y(t) = y0e6k

so obviously e6k = 1/2

Solve for k
 
  • #3
By the way, you don't really need to solve for k. Since
[tex]e^{kt}= (e^k)^t[/tex]
You really only need [itex]e^k[/tex].
 

1. What does the equation "Dy/dt=ky gallons of oil" represent?

The equation represents the rate of change of the amount of oil (y) in a given system over time (t), with the constant k representing the rate of change.

2. How is this equation used in the study of oil production?

This equation is commonly used in the study of oil production to predict the rate of oil production over time, taking into account factors such as depletion of resources and production techniques.

3. What does the variable "y" represent in this equation?

In this equation, "y" represents the amount of oil in gallons. It is the variable that is changing over time.

4. What is the significance of the constant "k" in this equation?

The constant "k" represents the rate of change in the amount of oil. It can be determined through various production factors such as well productivity, technology, and reservoir characteristics.

5. Can this equation be applied to other natural resources?

Yes, this equation can be applied to other natural resources such as gas or coal, as long as the rate of change and units are adjusted accordingly.

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