A composite solid comprises a frustum of a cone mounted on a hemisphere. Calculate the volume of the solid correct to two decimal places.

The diameter of the hemisphere is 30 cm.
The radius of the hemisphere ($R_H$) is half the diameter:

$$R_H = \frac{30}{2} = 15 \text{ cm}$$

The height of the hemisphere is equal to its radius:

$$h_H = 15 \text{ cm}$$

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

$$V_H = \frac{2}{3} \times \frac{22}{7} \times (15)^3$$ $$V_H = \frac{2}{3} \times \frac{22}{7} \times 3375$$ $$V_H = \frac{44}{21} \times 3375$$ $$V_H = \frac{148500}{21} = \frac{49500}{7} \text{ cm}^3$$

The radius of the bottom of the frustum ($R_B$) is equal to the radius of the hemisphere:

$$R_B = 15 \text{ cm}$$

The radius of the top of the frustum ($R_T$) is given as:

$$R_T = 10 \text{ cm}$$

The total height of the solid is 60 cm. The height of the frustum ($h_F$) is the total height minus the height of the hemisphere:

$$h_F = 60 - h_H = 60 - 15 = 45 \text{ cm}$$

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

$$V_F = \frac{1}{3} \times \frac{22}{7} \times 45 \times (15^2 + 15 \times 10 + 10^2)$$ $$V_F = \frac{1}{3} \times \frac{22}{7} \times 45 \times (225 + 150 + 100)$$ $$V_F = \frac{22}{7} \times 15 \times (475)$$ $$V_F = \frac{330}{7} \times 475$$ $$V_F = \frac{156750}{7} \text{ cm}^3$$

The total volume ($V_{total}$) is the sum of the volume of the hemisphere and the volume of the frustum.

$$V_{total} = V_H + V_F$$ $$V_{total} = \frac{49500}{7} + \frac{156750}{7}$$ $$V_{total} = \frac{49500 + 156750}{7}$$ $$V_{total} = \frac{206250}{7}$$ $$V_{total} \approx 29464.2857 \text{ cm}^3$$

Rounding to two decimal places:

$$V_{total} \approx \boxed{29464.29 \text{ cm}^3}$$

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The side of the cube ($s$) is 16 cm.
The volume of one cube ($V_{cube}$) is $s^3$.

$$V_{cube} = (16)^3$$ $$V_{cube} = 16 \times 16 \times 16$$ $$V_{cube} = 4096 \text{ cm}^3$$

To find the number of cubes, divide the total volume of the solid by the volume of one cube. We use the exact value of the total volume for accuracy before rounding.

$$\text{Number of cubes} = \frac{V_{total}}{V_{cube}}$$ $$\text{Number of cubes} = \frac{206250/7}{4096}$$ $$\text{Number of cubes} = \frac{206250}{7 \times 4096}$$ $$\text{Number of cubes} = \frac{206250}{28672}$$ $$\text{Number of cubes} \approx 7.1934$$

Since only whole cubes can be realized, we take the integer part of the result.

$$\text{Number of cubes} = \boxed{7}$$