Heroku Error R14 (Memory quota exceeded): How do I solve this?

I have a Rails 3.1 app on Heroku. I am seeing a lot of these errors:

Error R14 (Memory quota exceeded)

Typically the preceding log entry is showing:

Process running mem=522M(102.1%)

This does vary a little but never by much and can occur after almost any URL request, so it's not related to a specific controller action as far as I can tell.

This is a classic block of log entries:

2012-01-16T02:35:57+00:00 heroku[router]: PUT prizequiz.herokuapp.com/mobile_users/1 dyno=web.1 queue=0 wait=0ms service=55ms status=401 bytes=27
2012-01-16T02:35:58+00:00 heroku[router]: PUT prizequiz.herokuapp.com/mobile_users/1 dyno=web.1 queue=0 wait=0ms service=155ms status=200 bytes=1
2012-01-16T02:36:02+00:00 heroku[router]: PUT prizequiz.herokuapp.com/mobile_users/1 dyno=web.1 queue=0 wait=0ms service=13ms status=401 bytes=27
2012-01-16T02:36:02+00:00 heroku[router]: PUT prizequiz.herokuapp.com/mobile_users/1 dyno=web.1 queue=0 wait=0ms service=147ms status=200 bytes=1
2012-01-16T02:36:09+00:00 heroku[router]: POST prizequiz.herokuapp.com/mobile_users dyno=web.1 queue=0 wait=0ms service=87ms status=201 bytes=624
2012-01-16T02:36:11+00:00 heroku[router]: GET prizequiz.herokuapp.com/quizzes/1/questions dyno=web.1 queue=0 wait=0ms service=5ms status=401 bytes=27
2012-01-16T02:36:11+00:00 heroku[router]: GET prizequiz.herokuapp.com/quizzes/1/questions dyno=web.1 queue=0 wait=0ms service=290ms status=200 bytes=81412
2012-01-16T02:36:15+00:00 heroku[router]: PUT prizequiz.herokuapp.com/mobile_users/1 dyno=web.1 queue=0 wait=0ms service=10ms status=401 bytes=27
2012-01-16T02:36:16+00:00 heroku[router]: PUT prizequiz.herokuapp.com/mobile_users/1 dyno=web.1 queue=0 wait=0ms service=67ms status=200 bytes=1
2012-01-16T02:36:33+00:00 heroku[router]: POST prizequiz.herokuapp.com/quizzes/1/scores dyno=web.1 queue=0 wait=0ms service=10ms status=401 bytes=27
2012-01-16T02:36:33+00:00