Question

A right Circular Cone is $3.6\mathrm{cm}$ high and has base radius $1.6\mathrm{cm}$. It is melted and recast into a right circular cone with radius of its base as $1.2\mathrm{cm}$, find its height.

Answer 1

Step 1: Given information

The base radius of first right circular cone $\left(r_1\right)$ is $1.6\mathrm{cm}$.

The height of first right circular cone $\left(h_1\right)$ is $3.6\mathrm{cm}$.

The base radius of second right circular cone $\left(r_2\right)$ is $1.2\mathrm{cm}$.

Step 2: Determine the volume of first and second right circular cone

We know that, the volume of right circular cone is $\frac{1}{3}\mathrm{\pi }{\left(r\right)}^2\mathrm{h}$, where $r$ is base radius of right circular cone and $h$ is height of first right circular cone.

$\begin{array}{rcl}\mathrm{Volume}\mathrm{of}\mathrm{first}\mathrm{right}\mathrm{circular}\mathrm{cone}\left({\mathrm{V}}_1\right)& =& \frac{1}{3}\times \mathrm{\pi }\times {\left(1.6\right)}^2\times 3.6\\ & =& \frac{1}{3}\times \mathrm{\pi }\times 2.56\times 3.6\\ & =& \frac{9.216}{3}\mathrm{\pi }{\mathrm{cm}}^3\end{array}$

$\begin{array}{rcl}\mathrm{Volume}\mathrm{of}\mathrm{second}\mathrm{right}\mathrm{circular}\mathrm{cone}\left({\mathrm{V}}_2\right)& =& \frac{1}{3}\times \mathrm{\pi }\times {\left(1.2\right)}^2\times 3.6\\ & =& \frac{1}{3}\times \mathrm{\pi }\times 1.44\times h_2\\ & =& \frac{1.44}{3}\times \mathrm{\pi }\times h_2{\mathrm{cm}}^3\end{array}$

Step 3: Determine the height of second right circular cone after recasting

Since, second right circular cone is made from first right circular cone, therefore,

$\begin{array}{rcl}\mathrm{Volume}\mathrm{of}\mathrm{Second}\mathrm{Right}\mathrm{Circular}\mathrm{Cone}\left({\mathrm{V}}_2\right)& =& \mathrm{Volume}\mathrm{of}\mathrm{First}\mathrm{Right}\mathrm{Circular}\mathrm{Cone}\left({\mathrm{V}}_1\right)\\ \frac{1.44}{3}\times \mathrm{\pi }\times h_2& =& \frac{9.216}{3}\times \mathrm{\pi }\\ h_2& =& \frac{9.216}{1.44}\\ {\mathrm{h}}_2& =& 6.4\mathrm{cm}\end{array}$

Hence, the height of new right circular cone after recasting is $6.4\mathrm{cm}$.