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.

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Solution

## Equating volumes of both conesGiven 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. $⇒{V}_{1}={V}_{2}$ where ${V}_{1}$ is the volume of the original cone and ${V}_{2}$ is the volume of the recast cone.$⇒\frac{1}{3}\mathrm{\pi }{\left({\mathrm{r}}_{1}\right)}^{2}{\mathrm{h}}_{1}=\frac{1}{3}\mathrm{\pi }{\left({\mathrm{r}}_{2}\right)}^{2}{\mathrm{h}}_{2}$ (Volume of the cone $=\frac{1}{3}{\mathrm{\pi r}}^{2}\mathrm{h}$ )$⇒{\left(1.6\right)}^{2}×3.6={\left(1.2\right)}^{2}×{\mathrm{h}}_{2}$$⇒{\mathrm{h}}_{2}=\frac{{\left(1.6\right)}^{2}×3.6}{{\left(1.2\right)}^{2}}$$⇒{\mathrm{h}}_{2}=\frac{2.56×3.6}{1.44}$$⇒{\mathrm{h}}_{2}=\frac{9.216}{1.44}$$⇒{\mathrm{h}}_{2}=6.4cm$Hence, the height of the recast cone is $6.4cm$.

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