Interesting Quadratic Relationship - Tape Measurer

Spurred on by What can you do with this: Annuli by Dan Meyer,

I handed out tickets to the students as they entered my classroom on the first day of class in my Algebra II class, on each ticket, I write a number for their desk, that they have to find in the room.  After some of the obligatory first day stuff, I have them inquire about the roll of tickets and through a joint process we slowly come up with a plan of action on how to measure the ticket roll and estimate the number tickets.  The exercise goes fairly well, with the sole purpose to get them to start thinking of math as to answer questions of curiosity.   Anyhow as an extension, I was thinking about these big 30 meter tape measurers we have in our science department, and wondered if we could come up with a relationship between the number of turns it takes to roll in the tape and the reading on the tape measurer.   We collected the data a few days ago, but haven't done anything with the data yet.  However, I took a sneak peak  at the data and it provides a very strong quadratic relationship.

I plan to have the students plot the data and see if they can developed a quadratic function from data by choosing 3 points and solving a system of equations.  Obviously not quite as accurate as doing a full blown quadratic regression on the calculator, but hopefully it give them a feel for solving a system of equations as well as being able to make some predictions with their data.  Anyhow those are my thoughts at the moment.

In addition, I think I would like them to think about WHY this is a quadratic relation.  I'm not sure how well that exploration will go, but I am curious what they think.  Comments???