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Velocity/Net-Change type word problem

Here is the problem (51):

Attachment 28372

I've gotten as far as finding the function that gives the future amount of oil reserves:

Q(t)=(2*10^{9}) + 10^{7}((1 - e^{-kt})/k)

I have tried working out Q(1), but solving for k is still proving difficult. Any hints?

Re: Velocity/Net-Change type word problem

Check your signs - the formula should have Q(t) decreasing with time, not increasing. You should have:

$\displaystyle Q(t) = Q_0 +\frac {R_0} k (e^{-kt} -1)$

If you set t to infinity you get:

$\displaystyle Q(\infty) = Q_0 + \frac {R_0} k (e^{-\infty} -1) = Q_0 - \frac {R_0} k $

Set $\displaystyle Q(\infty) = 0$ to determine the minimum value of k in terms of $\displaystyle Q_0$ and $\displaystyle R_0$ that satisfies the condition.