Let Down Your Hair

Rapunzel. She had long hair. Really long. So long, that although locked in a room at the top of a stairless tower (by the pettiest neighbor-lady ever), she could still entertain the occasional visitor by dropping her hair out the window for them to climb (one assumes hand-over-hand, gym class style). Both in classic stories and modern adaptations like Disney’s Tangled, Rapunzel’s exceptionally long hair is taken as a given. Nobody says, “Hey, how does this woman even have hair that is as long as the height of a prison tower? Everyone knows an ambitious high school junior whose hair falls past her butt, but I’ve never seen anything like Rapunzel’s mane on a real person’s head. Not even close. Is there any way this situation even makes sense?”

Enter Ida Pederson. Ida is an engineer who lives in Sweden. She writes a blog with exhaustive information about healthy hair, growing your hair really long, and taking care of long hair. Part of the story is that she let her hair grow for ten years without cutting it, and luckily for us, took many selfies along the way.

From Ida’s photo documentation, we can glean many data points. For many moments along her decade-long journey, we can see and record the length of her hair. Plotting all these points (length of hair vs month) in the first quadrant allows us to see the linear trend of her hair’s growth, and estimate or compute a best-fit line representing the length of her hair in inches in terms of how many months it had been growing.

The slope of this line, helpfully, tells us the overall rate of Ida’s hair growth in inches per month. Which turns out to be about half an inch. Another way to estimate her hair growth rate is to observe that after ten years, Ida’s hair was around around 60 inches long, and 60 inches divided by 120 months gives us 0.5 inch per month.

By way of comparison, this part of the trailer for Tangled gives a sense of how long Disney animators imagined Rapunzel’s hair to be.

Let’s call it 40 feet long.

To reach a length of 40’ would, at a rate of ½” per month, require 80 years of hair growing. However, the film puts her age at 18. So either Rapunzel is lying on her driver’s license, by a lot, or her hair grows at an insane clip. But just how insane?

Let’s assume that Rapunzel’s hair, throughout her life, obeys the rudimentary characteristics of human hair: it started out nonexistent or very short when she was a baby, it grows at a more-or-less steady rate, and it would only get shorter if someone were to cut it. Let’s further assume it had never been cut before the time we meet the maiden in the tower.

Using our simplifying assumptions, that would mean in order for her hair to grow 40’ in 18 years, it would have to grow at a rate of around 2¼ feet a year, or around 2¼ inches a month (since there are as many months in a year as inches in a foot).

The implication is clear: enchanted, fairy-tale hair.

Math teachers: want to have this conversation with your students? Check out this week’s featured lesson, Let Down Your Hair.

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