What is the volume of a compound solid with a hemisphere and a cylindrical base?

The volume of a compound solid with a hemisphere and a cylindrical base is the sum of their volumes.

To find the volume of this compound solid, you need to calculate the volume of the hemisphere and the volume of the cylinder separately, then add them together. The formula for the volume of a hemisphere is \(\frac{2}{3} \pi r^3\), where \(r\) is the radius. The formula for the volume of a cylinder is \(\pi r^2 h\), where \(r\) is the radius and \(h\) is the height of the cylinder.

First, let's calculate the volume of the hemisphere. If the radius of the hemisphere is \(r\), then its volume is \(\frac{2}{3} \pi r^3\). This formula comes from the fact that a hemisphere is half of a sphere, and the volume of a sphere is \(\frac{4}{3} \pi r^3\).

Next, calculate the volume of the cylindrical base. If the radius of the cylinder is the same as the hemisphere, \(r\), and the height of the cylinder is \(h\), then its volume is \(\pi r^2 h\). This formula is derived from the area of the circular base, \(\pi r^2\), multiplied by the height, \(h\).

Finally, add the two volumes together to get the total volume of the compound solid. So, the total volume \(V\) is given by:
\[ V = \frac{2}{3} \pi r^3 + \pi r^2 h \]

By substituting the values of \(r\) and \(h\) into this formula, you can find the volume of the compound solid.