Does it bug anyone else not to have the analytic solution?
Fun story. I was in college taking differential equations. And I was faithfully attempting to apply what I'd learned in class, and having a somewhat hard time at it (this was before I realized how useful matlab was). When I was doing some research on the side, and I learned that the vast majority of differential equations are unsolvable, you can only approximate an answer. After that, I was thouroughly annoyed at wasting time in a class doing all this math by hand, and having to assume the answer was of a form that was solvable (and, yes, many useful engineering diff eq are of a solvable form).
So, now to answer your question. I thought this backboard seemed like a problem that dish makers should be able to trivially solve, but getting to know the math, and modifying it enough to solve this particular problem can take more time than setting up a simple probabilistic simulation. And after my experience with diffeq, I'm happy with a quick and dirty approximate solution, and then moving on with your life. Then maybe you run into someone who knows the math and is motivated enough to apply it, and maybe you update your project then.