To determine the volume of a compound solid that consists of a hemisphere atop a cylindrical base, we will calculate the volumes of both components separately and then combine them.

Volume Formulas

1. Volume of the Hemisphere: The formula for the volume of a hemisphere is given by:

   $$V_{hemisphere} = \frac{2}{3} \pi r^3$$

   Here, $r$ represents the radius of the hemisphere. This formula is derived from the fact that a hemisphere is half of a sphere, whose volume is calculated using the formula $\frac{4}{3} \pi r^3$.

2. Volume of the Cylinder: The volume of the cylindrical base can be calculated using the formula:

   $$V_{cylinder} = \pi r^2 h$$

   In this equation, $r$ is the radius of the cylinder (which is the same as the hemisphere), and $h$ is the height of the cylinder. This formula is based on the area of the circular base, $\pi r^2$, multiplied by the height $h$.

Total Volume Calculation

To find the total volume $V$ of the compound solid, we simply add the volumes of the hemisphere and the cylinder together:

Conclusion

By substituting the specific values for $r$ and $h$ into the formula above, you can easily calculate the total volume of the compound solid.