A right circular cone is enveloping a right circular cylinder that rests on the base of the cone. If the radius and height of the cone is $4 cm$ and $10 cm$ respectively, and the radius of cylinder is $\text{'}r\text{'} cm$, the largest possible curved surface area of the cylinder is $\text{'}a\mathrm{\pi r}\left(\mathrm{b}-\mathrm{r}\right)\text{'}$. Then $a\times b=?$

A vessel is in the form of a hollow cylinder mounted on a hemispherical bowl. The diameter of the sphere is $14 cm$ and the total height of the vessel is $13 cm$. Find the capacity of the vessel. (Take $\mathrm{\pi }=\frac{22}{7}$)

The sides of a triangle are $21,20 \text{and} 13 cm$. Find the area of the larger triangle into which the given triangle is divided by the perpendicular upon the longest side from the opposite vertex.

A circular tent is cylindrical to a height of $3$ meters and conical above it. If its diameter is $105 m$ and the slant height of conical portion is $53 m$, calculate the length of the canvas $5 m$ wide to make the required tent.

A steel sphere of radius $4 cm$ is drawn into a wire of diameter $4 mm$. Find the length of wire.

A cylinder and a cone having equal diameter of their bases are placed in the Qutub Minar one on the other, with the cylinder placed in the bottom. If their curved surface area are in the ratio of $8:5$, find the ratio of their heights. Assume the height of the cylinder to be equal to the radius of Qutub MInar.( Assume Qutub Minar to be having same radius throughout).

If the curved surface area of a cone is thrice that of another cone and slant height of the second cone is thrice that of the first, find the ratio of the area of their base.