Question

The outer curved surface areas of hemisphere and sphere are in the ratio 2:9. Find the ratio of their radii.

Answer 1

Dear Student,
$$\begin{gathered} CSAofhemisphere=2{{\mathrm{\pi r}}_1}^2 \\ \mathrm{CSA}\mathrm{of}\mathrm{sphere}=4{{\mathrm{\pi r}}_2}^2 \\ \mathrm{Now}\mathrm{given} \\ \frac{\mathrm{CSA}\mathrm{of}\mathrm{hemisphere}}{\mathrm{CSA}\mathrm{of}\mathrm{Sphere}}=\frac{2}{9} \\ \frac{2{{\mathrm{\pi r}}_1}^2}{4{{\mathrm{\pi r}}_2}^2}=\frac{2}{9} \\ \frac{{{\mathrm{r}}_1}^2}{2{{\mathrm{r}}_2}^2}=\frac{2}{9} \\ \frac{{{\mathrm{r}}_1}^2}{{{\mathrm{r}}_2}^2}=\frac{4}{9} \\ \mathrm{Taking}\mathrm{Square}\mathrm{root}\mathrm{both}\mathrm{side} \\ \mathrm{we}\mathrm{get}, \\ \sqrt{\frac{{{\mathrm{r}}_1}^2}{{{\mathrm{r}}_2}^2}}=\sqrt{\frac{4}{9}} \\ \frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}=\frac{2}{3} \\ \mathrm{Hence}{\mathbf{r}}_{\mathbf{1}}\mathbf{:}{\mathbf{r}}_{\mathbf{2}}\mathbf{=}\mathbf{2}\mathbf{:}\mathbf{3} \end{gathered}$$
Regards