Question

Two solid spheres of radii $2\text{cm}$ and $4\text{cm}$ are melted and recast into a cone of height $8\text{cm}$. Find the radius of the cone (in $\text{cm}$) so formed.

Answer 1

The volume of solid sphere of radius $2\text{cm}=\frac{4}{3}\pi \times 2^3$ $[\because$ volume of sphere $=\frac{4}{3}\pi \times r^3]$

The volume of solid sphere of radius $4\text{cm}=\frac{4}{3}\pi \times 4^3$ $[\because$ volume of sphere $=\frac{4}{3}\pi \times r^3]$

Volume of cone $=\frac{1}{3}\pi \times r^2\times h$

Now, as per the question

Volume of two spheres $=$ Volume of cone
$\frac{4}{3}\pi {\left(2\right)}^3+\frac{4}{3}\pi {\left(4\right)}^3=\frac{1}{3}\pi r^{2}h$
$32+256=r^2\times 8$
$\frac{288}{8}=r^2$
$36=r^2$
$r=6\text{cm}$.