Rice Dumplings!

Classical Mechanics Level 5

Rice dumpling, colloquially known as zongzi, is a traditional Chinese food cooked and enjoyed during Dragon Boat Festival. (You can read more about its rich cultural history here .) As shown, it has the shape of a regular tetrahedron , which has 4 faces.

Now, one of the faces of a zongzi filled evenly inside is chosen and a slice is made parallel to that face, passing through its center of mass .

In what ratio does the slice divide the rice dumpling?

This is part of the set Things Get Harder! .

1:8

1:7

8:27

8:19

27:64

27:37

1 solution

Donglin Loo
Jun 19, 2018

Relevant wiki: Barycentric coordinates

You can choose to read @Kelvin Hong 's blog $\textbf{Algebrot}$ on the same topic mentioned above here to get a clearer picture of what I am discussing about.

Let the vertices of the tetrahedron be $A,B,C,P$ .

Since the mass is evenly distributed, we may say the the four vertices each has a mass weightage of $1$ .

Let the chosen face contains the vertices $A,B,C$ .

$D,F$ are midpoints of $BC,AB$ respectively.

$B,C$ each has mass weightage of $1$ .

$\therefore$ mass weightage at $D$ is $2$ .

$A$ has mass weightage of $1$ .

You can view $AD$ as a lever with $E$ acting as fulcrum.

$\therefore AE:ED=2:1$

$\textbf{Note:}$ Point $E$ is the centroid of $\Delta ABC$ .

$A,D$ has mass weightage of $1,2$ respectively.

$\therefore$ mass weightage at $E$ is $3$ .

$P$ has mass weightage of $1$ .

Similiarly, we deduce that $PW:WE=3:1$

$\Rightarrow PW:PE=3:4$

$\textbf{Note:}$ Point $W$ is the centroid of $\Delta PAD$ and centre of mass of tetrahedron $ABCP$ .

The slice made produces a smaller tetrahedron similiar to the larger one.

The ratio of volume of the small tetrahedron to the large tetrahedron $=(\cfrac{3}{4})^3=\cfrac{27}{64}$

$\therefore$ The slice divides the rice dumpling in the ratio of $\cfrac{\cfrac{27}{64}}{1-\cfrac{27}{64}}=\cfrac{27}{37}$

I also do it with barycentric. As this is the 3-dimension aspect of barycentric, It makes this question really easy! Note that we can do it faster by only find out the ratio of zongzi's height. By barycentric, since the zongzi's mass is distributed evenly, so the mass on every verticle is the same, let's assume that be $1$ . So the bottom triangle plane has totally $3$ weight and top verticle has $1$ weight, so the center of mass will cut the zongzi into two parts where bottom part uses a quarter of the height and top part use three-quarter of the height. Finally, it can easily be solved by the similar figure (or the similar solid?) technique.

Kelvin Hong - 2 years, 11 months ago

Yeah, you are right. We can just sum three weight of mass on the bottom plane, thus giving us 3 to 1 mass ratio.

donglin loo - 2 years, 11 months ago

Rice Dumplings!

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