Question

A cone is $8.4cm$ high and the radius of its base is $2.1cm$. It is melted and recast into a sphere. The radius of the sphere is:

• A. $4.2cm$
• B. $2.1cm$
• C. $2.4cm$
• D. $1.6cm$

Answer 1

The correct option is B

$2.1cm$

Given, the height of cone, $h=8.4cm$.

The radius of base of the cone, $r=2.1cm$.

We know that, when a shape is melted and recast into another shape, then the volume of both the shapes becomes equal.

Let the radius of the sphere be $Rcm$. then

$\begin{array}{rcl}Volumeofcone& =& Volumeofsphere\\ \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}& =& \frac{4}{3}{\mathrm{\pi R}}^3\\ {\left(2.1\right)}^2\times 8.4& =& 4\times {\mathrm{R}}^3\\ {\mathrm{R}}^3& =& \frac{2.1\times 2.1\times 8.4}{4}\\ \mathrm{R}& =& \sqrt[3]{2.1\times 2.1\times 2.1}\\ \mathrm{R}& =& 2.1\mathrm{cm}\end{array}$

Hence, option B is correct.