Question

the ratio of volumes of two spheres is 64:8. find the ratio of their surface areas.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{R}\mathrm{be}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{sphere}\mathrm{and}\mathrm{r}\mathrm{be}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{second}\mathrm{sphere}. \\ \frac{\mathrm{volume}\mathrm{of}\mathrm{first}\mathrm{sphere}}{\mathrm{volume}\mathrm{of}\mathrm{second}\mathrm{sphere}}=\frac{64}{8} \\ \Rightarrow \frac{{\displaystyle \frac{4}{3}}{\mathrm{\pi R}}^3}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}=8 \\ \Rightarrow {\left(\frac{\mathrm{R}}{\mathrm{r}}\right)}^3=\frac{8}{1} \\ \Rightarrow \frac{\mathrm{R}}{\mathrm{r}}=\frac{2}{1} \\ \mathrm{Now},\frac{\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{first}\mathrm{sphere}}{\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{second}\mathrm{sphere}}=\frac{4{\mathrm{\pi R}}^2}{4{\mathrm{\pi r}}^2}={\left(\frac{\mathrm{R}}{\mathrm{r}}\right)}^2={\left(\frac{2}{1}\right)}^2=\frac{4}{1} \end{gathered}$$