Question

The radii of two spheres are in the ratio 2:3 find the ratio of their surface areas and ratio of their volumes.

Answer 1

$$\begin{gathered} \mathrm{Let}{\mathrm{r}}_1\mathrm{and}{\mathrm{r}}_2\mathrm{be}\mathrm{the}\mathrm{radii}\mathrm{of}2\mathrm{spheres}. \\ \mathrm{Now},\frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}=\frac{2}{3} \\ \mathrm{Now},\frac{\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{first}\mathrm{sphere}}{\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{second}\mathrm{sphere}}=\frac{4{{\mathrm{\pi r}}_1}^2}{4{{\mathrm{\pi r}}_2}^2}={\left(\frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}\right)}^2={\left(\frac{2}{3}\right)}^2=\frac{4}{9} \\ \frac{\mathrm{volume}\mathrm{of}\mathrm{first}\mathrm{sphere}}{\mathrm{volume}\mathrm{of}\mathrm{second}\mathrm{sphere}}=\frac{{\displaystyle \frac{4}{3}}{{\mathrm{\pi r}}_1}^3}{{\displaystyle \frac{4}{3}}{{\mathrm{\pi r}}_2}^3}={\left(\frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}\right)}^3={\left(\frac{2}{3}\right)}^3=\frac{8}{27} \end{gathered}$$