Question

A solid metallic sphere of diameter $21\mathrm{cm}$ is melted and recasted into a number of smaller cones, each of diameter $7\mathrm{cm}$ and height $3\mathrm{cm}$. Find the number of cones so formed.

Answer 1

Step 1: Determine the volume of the metallic sphere.

The volume of the metallic sphere is calculated by $V=\frac{4}{3}{\mathrm{\pi r}}^3$ where $r$ represents the radius of the metallic sphere.

$V=\frac{4}{3}\times \pi \times {\left(\frac{21}{2}\right)}^3$

Step 2: Determine the volume of the cone.

The volume of the cone is calculated by $V\text{'}=\frac{1}{3}\pi {\left(r\text{'}\right)}^{2}h$ where $r\text{'}$ represents the radius of the cone and $h$ represents the height of the cone.

$V\text{'}=\frac{1}{3}\pi {\left(\frac{7}{2}\right)}^2\times 3$

Step 3: Determine the number of cone.

$\begin{array}{rcl}\mathrm{Number}\mathrm{of}\mathrm{cones}& =& \frac{\frac{4}{3}\times \mathrm{\pi }\times \frac{21}{2}\times \frac{21}{2}\times \frac{21}{2}}{\frac{1}{3}\times \mathrm{\pi }\times \frac{7}{2}\times \frac{7}{2}\times 3}\\ & =& 126\end{array}$

Hence, there are $126$ number of cones formed.