A solid ball is exactly fitted inside the cubical box of side $a.$ The volume of the ball is $\frac{4}{3}\mathrm{\pi }{a}^3.$

The capacity of a cylindrical vessel with a hemispherical portion raised upward at the bottom as shown in the figure is $\frac{\pi r^2}{3}\left[3h-2r\right].$

The curved surface area of a cone is $\pi l\left(r_1+r_2\right)$, where $l=\sqrt{h^2+{\left(r_1+r_2\right)}^2 }$, $r_1$ and $r_2$ are the radius of the two ends of the frustum and $h$ is the vertical height.

Three metallic solid cubes whose edges are $3 \mathrm{cm}, 4 \mathrm{cm}$ and $5 \mathrm{cm}$ are melted and formed into a single cube. Find the edge of the cube so formed.

How many shots having diameter $3 \mathrm{cm}$ can be made from a cuboidal lead solid of dimensions $9 \mathrm{cm}\times 11 \mathrm{cm}\times 12 \mathrm{cm}$?

A bucket is in the form of a frustum of a cone and holds $28.490 \mathrm{L}$ of water. The radius of the top and bottom are $28 \mathrm{cm}$ and $21 \mathrm{cm}$, respectively. Find the height of the bucket.

A cone of radius $8 \mathrm{cm}$ and height $12 \mathrm{cm}$ is divided into two parts by a plane through the mid-point of its axis parallel to its base. Find the ratio of the volumes of two parts.