Question

A solid cylinder is melted and cast into a cone of same radius. The heights of the cone and cylinder are in the ratio

(a) 9 : 1

(b) 1 : 9

(c) 3 : 1

(d) 1 : 3

Answer 1

Since the cylinder is re cast into a cone both their volumes should be equal.

So, let Volume of the cylinder = Volume of the cone

= V

It is also given that their base radii are the same.

So, let Radius of the cylinder = Radius of the cone

= r

Let the height of the cylinder and the cone be and respectively.

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as

Volume of cone =

The formula of the volume of a cylinder with base radius ‘r’ and vertical height ‘h’ is given as

Volume of cylinder =

So we have
$$\begin{gathered} \frac{Volumeofcone}{Volumeofcylinder}=\frac{{\displaystyle \frac{1}{3}}{\mathrm{\pi r}}^2{\mathrm{h}}_{\mathrm{cone}}}{\pi r^2{\mathrm{h}}_{cylinder}} \\ \Rightarrow \frac{V}{V}=\frac{{\displaystyle \frac{1}{3}{\mathrm{h}}_{cone}}}{{\mathrm{h}}_{cylinder}} \\ \Rightarrow 1=\frac{{\displaystyle \frac{1}{3}{\mathrm{h}}_{cone}}}{{\mathrm{h}}_{cylinder}} \end{gathered}$$

Hence the correct choice is option (c).