Question

A solid metallic sphere of diameter $28\mathrm{cm}$ is melted and recasted into a number of smaller cones, each of diameter $4\frac{2}{3}\mathrm{cm}$ and height $3\mathrm{cm}$. Find the number of cones so formed.

Answer 1

Step 1: Determine the number of cones.

$\begin{array}{rcl}\mathrm{Number}\mathrm{of}\mathrm{cones}& =& \frac{\mathrm{Volume}\mathrm{of}\mathrm{metallic}\mathrm{sphere}}{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cone}}\\ & =& \frac{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}{{\displaystyle \frac{1}{3}}\mathrm{\pi }{\left(\mathrm{r}\text{'}\right)}^2\mathrm{h}}\\ & =& \frac{4r^3}{{\left(\mathrm{r}\text{'}\right)}^2\mathrm{h}}\\ & =& \frac{4\times 14\times 14\times 14}{{\displaystyle \frac{4{\displaystyle \frac{2}{3}}}{2}}\times {\displaystyle \frac{4{\displaystyle \frac{2}{3}}}{2}}\times 3}\\ & =& 672\end{array}$

Hence, there are $672$ cones are formed.