Question

A solid metallic cylinder of radius $3.5\text{cm}$ and height $14\text{cm}$ is melted and recast into a number of small solid metallic balls, each of radius $\frac{7}{12}\text{cm}$. Find the number of balls so formed.

Answer 1

Step 1 : Calculation of the volume of a solid metallic cylinder that is melted.

The volume of a cylinder is given by,

$\mathit{V}\mathbf{=}\mathit{\pi }{\mathit{r}}^{\mathbf{2}}\mathit{h}$, where $V$ is the volume, $r$ is the radius, and $h$ is the height of a cylinder.

It is given that $r=3.5\text{cm and}h=14\text{cm}$. Let the volume of this cylinder be $V_1$,

$\begin{array}{rcl}\therefore V_1& =& \pi r^{2}h\\ & =& \pi {\left(3.5\right)}^2\left(14\right)\\ & =& \pi \times 12.25\times 14\\ & =& 171.5\pi {\text{cm}}^3\end{array}$

Step 2 : Calculation of the volume of the metallic balls made from the melted cylinder.

As it is a solid metallic ball then it will be of a spherical shape.

The volume of a sphere is given by,

$\mathit{V}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}}\mathit{\pi }{\mathit{r}}^{\mathbf{3}}$, where $V$ is the volume and $r$ is the radius of a sphere.

It is given that $r=\frac{7}{12}\text{cm}$ and let the volume of a metallic ball be $V_2$

$\begin{array}{rcl}\therefore V_2& =& \frac{4}{3}\pi r^3\\ & =& \frac{4}{3}\pi {\left(\frac{7}{12}\right)}^3\\ & =& \frac{4}{3}\times \frac{343}{1728}\pi \\ & =& \frac{114.34}{432}\pi \\ & =& 0.2646\pi {\text{cm}}^3\end{array}$

Step 3 : Calculation of the number of solid metallic balls

The number of metallic balls can be determined by dividing the $V_1$ by $V_2$. Denote the number of balls by $N$.

$\begin{array}{rcl}\therefore N& =& \frac{V_1}{V_2}\\ & =& \frac{171.5\times \cancel{\pi }}{0.2646\times \cancel{\pi }}\\ & =& \frac{171.5}{0.2646}\\ & =& 648.148\end{array}$

Final Answer :

Hence, the total number of metallic balls that can be formed is $648$.