A solid is a piece of metal in the shape of a right circular cone mounted on a solid hemisphere with the same radius. The radius of the hemisphere is $\frac{1}{3}$of the conical part's vertical height. If the radius of the cone's base is r, the volume of the metal is

In this problem, we are given a solid in the shape of a right circular cone mounted on a solid hemisphere of the same radius. The solid is made of metal, and we need to calculate its volume. As a result, the volume of the solid will be the sum of the volumes of the cone and the hemisphere. We need the height of the cone and the radius of the base of the cone to calculate the volume of a cone, and the radius to calculate the volume of a hemisphere. Since the radius of the cone's base is given as r and some information about the cone's height is given. So we'll use it to find what we need.

Volume of the solid = Volume of the hemispherical part + Volume of conical part

Volume of a hemisphere of radius ${\mathrm{r}}^{\text{'}}=\frac{2}{3}\pi {\mathrm{r}}^3$

Volume of a cone $=\frac{1}{3}\pi r^{2}h$ where r is the radius of the base of the cone and h is the height.

Given, $r=\frac{1}{3} \mathrm{h}\Rightarrow \mathrm{h}=3\mathrm{r}$

Hence,

Volume of the solid $=\left(\frac{2}{3}\pi r^3\right)+\left(\frac{1}{3}\times \pi \times r^2\times 3r\right)$

$=\frac{2}{3}\pi r^3+\pi r^3=\frac{5}{3}\pi r^3$