Long Throw Projector For Auditorium

[AVSJake]

Looking for a projector that is going in a large auditorium that has a throw distance of 110 feet to a 16x12 screen. What is the best solution for an under 10k budget? 

How many lumens are needed to maintain a quality image at that distance?

[AVSCraig]

Is there ambient light or is it a dark environment ? 

[AVSJake]

There will be ambient light in most instances that it is being used

thanks!