Question

A right circular cone is $3.6cm$ high and the radius of its base is $1.6cm$. It is melted and recast into a right circular cone with radius of its base as $1.2cm$. Find its height.

Answer 1

Equating volumes of both cones

Given that, height $\left(h_1\right)$ of the original cone is $3.6cm$ and its radius $\left(r_1\right)$ is $1.6cm$.

Also, after melting the original cone and recasting it, the radius $\left(r_2\right)$ of the cone becomes $1.2cm$

We know that the cone is melted and recast so, the volume of the cones would be equal.

$\Rightarrow V_1=V_2$ where $V_1$ is the volume of the original cone and $V_2$ is the volume of the recast cone.

$\Rightarrow \frac{1}{3}\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^2{\mathrm{h}}_1=\frac{1}{3}\mathrm{\pi }{\left({\mathrm{r}}_2\right)}^2{\mathrm{h}}_2$ (Volume of the cone $=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$ )

$\Rightarrow {\left(1.6\right)}^2\times 3.6={\left(1.2\right)}^2\times {\mathrm{h}}_2$

$\Rightarrow {\mathrm{h}}_2=\frac{{\left(1.6\right)}^2\times 3.6}{{\left(1.2\right)}^2}$

$\Rightarrow {\mathrm{h}}_2=\frac{2.56\times 3.6}{1.44}$

$\Rightarrow {\mathrm{h}}_2=\frac{9.216}{1.44}$

$\Rightarrow {\mathrm{h}}_2=6.4cm$

Hence, the height of the recast cone is $6.4cm$.