Question

The surface area of two hemispheres are in the ratio 8:27. What is the ratio of their radii?

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{r}\mathrm{be}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{first}\mathrm{hemisphere}\mathrm{and}\mathrm{R}\mathrm{be}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{second}\mathrm{hemisphere}. \\ \mathrm{Now},\frac{\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{first}\mathrm{hemisphere}}{\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{second}\mathrm{hemisphere}}=\frac{8}{27} \\ \Rightarrow \frac{{\displaystyle \frac{2}{3}}{\mathrm{\pi r}}^2}{{\displaystyle \frac{2}{3}}{\mathrm{\pi R}}^2}=\frac{8}{27} \\ \Rightarrow \frac{{\mathrm{r}}^2}{{\mathrm{R}}^2}=\frac{8}{27} \\ \Rightarrow \frac{\mathrm{r}}{\mathrm{R}}=\frac{2\sqrt{2}}{3\sqrt{3}} \end{gathered}$$