Thank you for your suggestion. The only reason I wouldn't do that is because Rackspace was very adamant about even the extension of the deadline (they would only agree to a 5-day extension). But it doesn't hurt to ask, though it may create a strained relationship between us.
It's their job to try and collect as much as they can. Even if your contact is friendly and sympathetic, if they think you can pay on better terms than you offer they will hammer you. On the flip side it is your job to be honest about your situation and get them to accept terms that you can actually meet. Other people have given better and more specific advice but the bottom line is they will take what they can get. No hard feelings. Business is business.
I worked as a collector at one point. You have incentives to get money collected. A collector will say everything they can to get you to pay (sometimes skirting the collection laws).
They know that you have a limited amount of money and the chance you actually paying decreases as time goes by.