course mth175

Oil is pumped continuously from a well at a rate proportional to the amount of the oil left in the well. Initially, there were 1 million barrels of oil in the well' six years later 500,000 barrels remain. At what rate was the amount of oil in the well decreasing when there were 600,000 barrels remaining.

When will will there be 50,000 barrels remaining?

So to make a differential equation to model this we would we need to satisfy the fact that the rate at which oil is pumped from a well is proportional to the amount of oil left in the well

so would it follow that dO/dt=kr for the first step in the equation. I've worked through about 5 or 6 of these problems in 11.5 and had fairly good success. I'm a little unsure on whether or not you would need to model this on the rate oil leaves the well= rate of outflow * concentration?

The main thing is setting it up correctly.

O isn't a good thing to use for a variable, simply because it's easy to confuse with zero. However in terms of O = amt of oil remaining the equation would be

dO/dt = k O.

k r doesn't make sense because you haven't defined r.

Using Q (for quantity of oil) rather that O the equation would be

dQ/dt = k Q.

What is the general solution to this equation, and what do you get when you plug the given conditions into this solution?