The length scale factor between the smaller cylinder and the larger cylinder is 2, so the volume scale fator between the smaller cylinder and the larger cylinder is 2$^3$=8.

So the smaller cylinder will contain $\frac{1}{9}$ of the total volume, and the larger cylinder will contain $\frac{8}{9}$ of the total volume.

The volume of the cube is $9^3$ cm$^3$, so the volume of the smaller cylinder is $\frac{1}{9}\times9^3=81$ cm$^3$.

Suppose the smaller cylinder has height $h$ cm, so the larger cylinder has height $2h$ cm. Then the volume of the smaller cylinder is $\pi\times2^2\times h=4\pi h$ cm$^3$ and the volume of the larger cylinder is $\pi\times4^2\times2h=32\pi h$ cm$^3$.

The volumes of the cylinders will have to add up to the volume of the cube, so $$\begin{align}4\pi h+32\pi h=&9^3\\

\Rightarrow 36\pi h=&729\\

\Rightarrow h=&\frac{729}{36\pi}=\frac{81}{4\pi}\end{align}$$

Substituting into the expression for the volume of the smaller cylinder gives $4\pi h=4\pi\times\dfrac{81}{4\pi}=81$. So the volume of the smaller cylinder is $81$ cm$^3$.

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