Boy Volcanoes

I met Nichol and her two sons on the beach yesterday.

The elder boy decided he wanted to be buried and have a fake mermaid tail sculpted over his legs. So he lay down and we proceeded to pile sand on top of him. As I recall we didn’t do a very good job on the tail, but we did cover him.

But we didn’t put enough sand on him, and he was up again soon enough and into the water.

Then the younger son decided that he wanted to be buried too. I couldn’t quite figure out what he intended as the first thing he wanted was a narrow and fairly deep hole. And then another. It turned out these were for his legs, and he planted a foot in each hole (about knee high or so) and stood.

He wanted to be buried standing up.

I don’t think it had ever occurred to me that this was an option for burying someone, but, on the other hand, it would immobilize him in a way that a conventional sand burial would not. So I set to work and soon had the sand up to his shoulders.

Just as I’d gotten him trapped, his brother came up and decided that he too wanted to be buried.

Both boys want to be immobilized? (They tend to run around and fight each other, it’s very distracting, but now they both want to be forced to stand in one place? I couldn’t resist.)

The elder brother is taller than his sibling. I hadn’t quite realized what that meant. Now the volume of a cone is (π × r2 × h)/3. And the radius of a pile of sand is related to its height by the angle of repose. h = r × tan(θ). Wikipedia says the angle of repose for sand is 34°, while I measure a value of 32° on the photograph below (close enough). Anyway this means that the volume of of sand that I need to pile around a boy is proportional to the cube of his height (π × h3)/( 3 × tan(θ)2) = (π × h3)/1.2.

I need to add a lot more sand to the taller boy.

Luckily he also wants a hole for his feet (he only gets one hole containing both feet while his brother got two, one for each foot). But it takes forever to cover him up to his shoulders.

Unfortunately, before I finish, the younger brother gets bored and his volcano erupts and spews forth a boy. I have failed in my diabolical ploy to immobilize both of them.

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