What is the volume of a compound solid made of a sphere and a cone?

The volume of a compound solid made of a sphere and a cone is the sum of their individual volumes.

To find the volume of a compound solid made of a sphere and a cone, you need to calculate the volume of each shape separately and then add them together. The formula for the volume of a sphere is \(\frac{4}{3} \pi r^3\), where \(r\) is the radius of the sphere. The formula for the volume of a cone is \(\frac{1}{3} \pi r^2 h\), where \(r\) is the radius of the base of the cone and \(h\) is the height of the cone.

Let's break it down with an example. Suppose you have a sphere with a radius of 3 cm and a cone with a radius of 3 cm and a height of 5 cm. First, calculate the volume of the sphere:

\[ \text{Volume of the sphere} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \times 27 = 36 \pi \, \text{cm}^3 \]

Next, calculate the volume of the cone:

\[ \text{Volume of the cone} = \frac{1}{3} \pi (3)^2 (5) = \frac{1}{3} \pi \times 9 \times 5 = 15 \pi \, \text{cm}^3 \]

Finally, add the two volumes together to get the total volume of the compound solid:

\[ \text{Total volume} = 36 \pi + 15 \pi = 51 \pi \, \text{cm}^3 \]

So, the volume of the compound solid is \(51 \pi \, \text{cm}^3\). Remember to always use the same units for all measurements and to include \(\pi\) in your final answer unless you are asked to use a specific numerical value for \(\pi\).