The height of Cylinder A is equal to its diameter. The height and diameter of Cylinder B are each twice those of Cylinder A. The height of Cylinder $\mathrm{C}$ is equal to its diameter. The volume of Cylinder C is the sum of the volumes of Cylinders A and B. What is the ratio of the diameter of Cylinder $\mathrm{C}$ to the diameter of Cylinder A?
Suppose that the height, radius and diameter of Cylinder A are $h, r$ and $d$, respectively.

Note that $d=h$ and $d=2 r$ and so $h=2 r$.

The volume of Cylinder A is $\pi r^{2} h$ which equals $2 \pi r^{3}$.

Since the height and diameter of Cylinder B are twice that of Cylinder A, then these are $2 h$ and $2 d$, respectively, which means that the radius of Cylinder B is $2 r$ and the height of Cylinder B equals $4 r$.

The volume of Cylinder B is $\pi(2 r)^{2}(4 r)$ which equals $16 \pi r^{3}$.

Suppose that the height, radius and diameter of Cylinder $\mathrm{C}$ are $H, R$ and $D$, respectively.

Note that $D=H$ and $D=2 R$ and so $H=2 R$.

The volume of Cylinder $\mathrm{C}$ is $\pi R^{2} H$ which equals $2 \pi R^{3}$.

Since the sum of the volumes of Cylinders A and B equals the that of Cylinder C, then we obtain $2 \pi r^{3}+16 \pi r^{3}=2 \pi R^{3}$ and so $18 \pi r^{3}=2 \pi R^{3}$ or $R^{3}=9 r^{3}$.

This means that $R=\sqrt[3]{9} r$ and so $2 R=\sqrt[3]{9}(2 r)$ which gives $D=\sqrt[3]{9} d$.

Therefore, the ratio of the diameter of Cylinder $\mathrm{C}$ to the diameter of Cylinder A is $\sqrt[3]{9}: 1$.

ANSWER: $\sqrt[3]{9}: 1$