Question

THE DIAMETER OF THE EARTH IS APPROXIMATELY 4 TIMES THE DIAMETER OF THE MOON. FIND THE RATIO OF THEIR SURFACE AREAS.

Answer 1

$$\begin{gathered} \mathrm{Let}{\mathrm{r}}_1\mathrm{and}{\mathrm{d}}_1\mathrm{be}\mathrm{the}\mathrm{radius}\mathrm{and}\mathrm{diameter}\mathrm{of}\mathrm{the}\mathrm{Moon}. \\ \mathrm{Let}{\mathrm{r}}_2\mathrm{and}{\mathrm{d}}_2\mathrm{be}\mathrm{the}\mathrm{radius}\mathrm{and}\mathrm{diameter}\mathrm{of}\mathrm{the}\mathrm{Earth}. \\ \mathrm{Now},{\mathrm{d}}_2=4{\mathrm{d}}_1 \\ \Rightarrow 2{\mathrm{r}}_2=8{\mathrm{r}}_1\left[\mathbf{as}\mathbf{,}\mathbf{}\mathbf{diameter}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{\times }\mathbf{radius}\right] \\ \Rightarrow {\mathrm{r}}_2=4{\mathrm{r}}_1 \\ \mathrm{Now},\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{Moon}=4{{\mathrm{\pi r}}_1}^2 \\ \mathrm{Surface}\mathrm{area}\mathrm{of}\mathrm{earth}=4{{\mathrm{\pi r}}_2}^2 \\ \mathrm{Now},\frac{\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{Moon}}{\mathrm{Surface}\mathrm{area}\mathrm{of}\mathrm{Earth}}=\frac{4{{\mathrm{\pi r}}_1}^2}{4{{\mathrm{\pi r}}_2}^2}={\left(\frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}\right)}^2={\left(\frac{{\mathrm{r}}_1}{4{\mathrm{r}}_1}\right)}^2=\frac{1}{16} \end{gathered}$$