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Order of Magnitude Parenting: Lines at Legoland

The Technic Project X roller coaster at Legoland California
Yesterday we took the boys to Legoland, down near San Diego. We've been in Southern California four years, but we've never visited the Mecca of my childhood religion! Oh, plastic, interlocking blocks. Thank you for keeping me out of the heat of St. Louis summers while enriching my mind.

While waiting in line with Owen to ride one of the roller coasters, a kid in front of us asked his father, "How much longer do we have to wait in line?" The father responded saying it would take about 30 minutes. After hearing this exchange, Owen looked up at me as if to ask, "Seriously? 30 minutes?" So I decided to turn it into a teaching moment. "Let's figure it out!"

Owen timing the arrival of each roller coaster car. The cars turned left over our heads and dropped to the left of this picture. We estimated 40 seconds per car. 

The first thing we did was look up above the line area, where the coaster was dragged to the top of the "hill" before plunging nearly straight down, right by the waiting patrons. We timed how long it took for consecutive cars to drop: 41 seconds. Let's call it a car 40 seconds. This comes to 1.5 cars per minute. Each car has 4 seats, so that's 10 people per minute passing overhead. Since there's a steady-state flow of people from the line to the point where they are dropped, screaming over the precipice, This is the rate, $R$, at which people are moving in the line.

Three different sub-lines within the main line. About 20 people per sub-line.

Next, we estimated how many people were in the line. I held Owen up and he counted the number of people in our "sub-line," where the main line wound back on itself. There were 5 of these sub-lines, with about 20 people per sub-line, for a total of $N = 100$ people in line. Thus

$T_{\rm left\ in\ line} = N / R = 100/10 = 10$ minutes

Owen was much happier with this estimate, as was I. As an added bonus, we got to pass the time doing a practical OoM problem.

In the end, it took us 13 minutes instead of 10. After the adrenaline of the ride wore off (it took a while, since it was Owen's first roller coaster), I asked him if he could think of why it took longer. He first answered that it was because we counted wrong. Good point. But I don't think were off by 30% just because of a counting error. In the end, we decided that the cars must be only partially full, on average, with less than 4 riders each.

We went back for a second ride on the roller coaster, and sure enough, every third car or so had only 3 riders. Families and other groups with 3, 5, 7, or any other odd number of riders would result in a partially filled car. In the final analysis, I figured that our counting error was about 10%, and the car fill factor was only about 85%. The second time through the line I estimated our time from the end of the line (6.5 sub-lines). After accounting for the fill-factor (correction of 1.2), I estimated 15 +/- 1 minutes. We got to the front in 14 minutes. Nailed it!

While I could be proud of my abilities of estimation, I was a little less proud of how I did on the ride, as captured by the coaster cam. I did a little bit better than Marcus, but with a lot less dignity:


Look closely and you can see Marcus in the back left seat. Owen's look is priceless. You wouldn't know it from this picture, but Erin was the one who wanted to skip the ride at boarding time.

Overall, it was a really enjoyable day!

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