Question

A piece of metal in the form of a cone of radius $3\mathrm{cm}$ and height $7\mathrm{cm}$ is melted and recast into a cube. find the side of the cube.

Answer 1

Step 1: Given information

The radius of the cone $\left(r\right)$ is $3\mathrm{cm}$.

The height of the cone $\left(h\right)$ is $7\mathrm{cm}$.

Step 2: Determine the volume of the cone

We know that, volume of the cone $\left(V_1\right)$ is $\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$, where $r$ is the radius of the cone and $h$ is the height of the cone.

$\begin{array}{rcl}\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cone}\left({\mathrm{V}}_1\right)& =& \frac{1}{3}\times \mathrm{\pi }\times {\mathrm{r}}^2\times \mathrm{h}\\ & =& \frac{1}{3}\times \frac{22}{7}\times {\left(3\right)}^2\times 7\\ & =& 22\times 3\\ & =& 66{\mathrm{cm}}^3\end{array}$

Step 3: Determine the side of the cube

We know that, volume of the cube is $s^3$, where $s$ is the side of the cube.

Since, Cube is made from cone, therefore,

$\begin{array}{rcl}\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{Cube}& =& \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{Cone}\\ {\mathrm{s}}^3& =& 66\\ \mathrm{s}& =& \sqrt[3]{66}\\ \mathrm{s}& =& 4.04\mathrm{cm}\end{array}$

Hence, the side of the cube is $4\mathrm{cm}$ approximately.