Calculate the volume of the composite solid correct to two decimal places.

The diameter of the hemisphere is 30 cm.
The radius of the hemisphere, $R_H = \frac{30}{2} = 15$ cm.

The formula for the volume of a hemisphere is $V_H = \frac{2}{3} \pi R_H^3$.
Using $\pi = \frac{22}{7}$:

The frustum is mounted on the hemisphere, so its bottom diameter is 30 cm.
The radius of the bottom of the frustum, $R_B = \frac{30}{2} = 15$ cm.
The radius of the top of the frustum, $R_T = 10$ cm.
The entire solid has a vertical height of 60 cm.
The height of the hemisphere is equal to its radius, $R_H = 15$ cm.
The height of the frustum, $h_F = \text{Total height} - \text{Height of hemisphere}$

The formula for the volume of a frustum is $V_F = \frac{1}{3} \pi h_F (R_B^2 + R_B R_T + R_T^2)$.
Using $\pi = \frac{22}{7}$:

The total volume $V_{total} = V_H + V_F$.

Rounding to two decimal places:
The volume of the solid is $\boxed{29464.29 \text{ cm}^3}$.

The side of the cube, $s = 16$ cm.
The volume of one cube, $V_C = s^3$.

Number of cubes = $\frac{\text{Total volume of solid}}{\text{Volume of one cube}}$

Since only whole cubes can be realized, we take the integer part.
The number of cubes realized is $\boxed{7}$.