The energy radiated per second by a black body is given by Stefan's law,

    $\frac{\text{E}}{\text{t}}=\sigma {\text{T}}^4\times \text{A}$

where A is the surface area of the black body

    $\frac{\text{E}}{\text{t}}=\sigma {\text{T}}^4\times 4\pi {\text{r}}^2$

Since black body is a sphere, $\text{A}=4\pi {\text{r}}^2$

Case (i)

       $\frac{\text{E}}{\text{t}}=450\text{, }\text{T}=500K\text{, }\text{r}=\text{0.12}m$

$\therefore 450=4\pi \sigma {\left(500\right)}^4{\left(\text{0.12}\right)}^2$                               ...(i)

Case (ii)

$\frac{\text{E}}{\text{t}}=\text{?}\text{, }\text{T}=1000K\text{, }\text{r}=0.06m$                      ...(ii)

Dividing Eq. (ii) by Eq. (i), we get

    $\frac{\text{E}/\text{t}}{450}=\frac{{\left(1000\right)}^4{\left(\text{0.06}\right)}^2}{{\left(500\right)}^4{\left(\text{0.12}\right)}^2}=\frac{2^4}{2^2}=4$

$\Rightarrow \frac{\text{E}}{\text{t}}=450\times 4=1800 W$