According to the question it is given that,

Radius of the copper sphere $=3cm$

As we know that,

Volume of the sphere $=4/3\pi {{r}^{3}}$

$=4/3\pi \times {{3}^{3}}$ ….. (i)

The copper sphere is melted and recasted into a right circular cone

Height of the cone $=3cm$

As we know that,

Volume of the right circular cone $=1/3\pi {{r}^{2}}h$

$=1/3\pi \times {{r}^{2}}\times 3$….. (ii)

On comparing equation (i) and (ii) we have,

$4/3\pi \times {{3}^{3}}=1/3\pi \times {{r}^{2}}\times 3$

${{r}^{2}}=36$

$r=6cm$

Therefore, the radius of the base of the cone is $6cm$.