# Compounding interest formula going awry

1. Oct 1, 2012

### Locoism

If Po dollars are deposited in an account paying r percent compounded continuously and withdrawals are at a rate of 200t (continuously), what is the amount after T years?

I derived the formula by taking the limit as m -> ∞ of the compounding interest equation P(t) = Po(1+r/m)^mt which gives us P(t) = Po*e^rt. So including the withdrawal rate

- P(t) = Po * e^rt - 200t

And that would be the amount in the account after T years. But my question asks, if r = .1 and Po = $5000, when will the account be empty? My function doesn't cross zero, and I don't understand where I made the mistake. Can anyone help me? 2. Oct 1, 2012 ### Mute If withdrawals are done at a rate of 200t, then isn't the total amount of money withdrawn between time 0 and t 100t2? 3. Oct 1, 2012 ### Locoism Sorry, the question says withdrawals are at an annual rate of 200t dollars. I thought about that, tried it out and the question still doesn't work out. I'm really worried about this one... *edit* Actually, I am an idiot. For some reason I got no real roots the first time... *edit* However the question asks me to consider the case Po =$20 000 and r=0.1. I don't see anything special about this. Intuitively I would say the amount stays constant, but plotting it gives me something completely different...

Thank you for your support :P

Last edited: Oct 1, 2012
4. Oct 4, 2012

### Staff: Mentor

The rate at which interest is accumulating is Pr, where r is the yearly interest rate divided by 100. The rate at which withdrawals are being made is 200. Therefore, the net rate at which principal is increasing is given by the differential equation:

dP/dt = Pr - 200