Question

The diameters of two cones are in the ratio 5:4 and their slant heights are in the ratio 3:8. Find the ratio of their curved surface areas.

Answer 1

$$\begin{gathered} \mathrm{Let}{\mathrm{d}}_1\mathrm{be}\mathrm{the}\mathrm{diameter}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{cone}\mathrm{and}{\mathrm{d}}_2\mathrm{be}\mathrm{the}\mathrm{diameter}\mathrm{of}\mathrm{the}\mathrm{second}\mathrm{cone}. \\ \mathrm{Now},\frac{{\mathrm{d}}_1}{{\mathrm{d}}_2}=\frac{5}{4} \\ \Rightarrow \frac{2{\mathrm{r}}_1}{2{\mathrm{r}}_2}=\frac{5}{4} \\ \Rightarrow \frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}=\frac{5}{4}\left[\mathrm{where}{\mathrm{r}}_1=\mathrm{radius}\mathrm{of}\mathrm{first}\mathrm{cone}\mathrm{and}{\mathrm{r}}_2=\mathrm{radius}\mathrm{of}\mathrm{second}\mathrm{cone}\right] \\ \mathrm{Now},\frac{\mathrm{slant}\mathrm{height}\mathrm{of}\mathrm{first}\mathrm{cone}}{\mathrm{slant}\mathrm{height}\mathrm{of}\mathrm{second}\mathrm{cone}}=\frac{3}{8} \\ \Rightarrow \frac{{\mathrm{l}}_1}{{\mathrm{l}}_2}=\frac{3}{8} \\ \mathrm{Now}, \\ \frac{\mathrm{CSA}\mathrm{of}\mathrm{first}\mathrm{cone}}{\mathrm{CSA}\mathrm{of}\mathrm{second}\mathrm{cone}}=\frac{{\mathrm{\pi r}}_1{\mathrm{l}}_1}{{\mathrm{\pi r}}_2{\mathrm{l}}_2}=\frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}\times \frac{{\mathrm{l}}_1}{{\mathrm{l}}_2} \\ \Rightarrow \frac{\mathrm{CSA}\mathrm{of}\mathrm{first}\mathrm{cone}}{\mathrm{CSA}\mathrm{of}\mathrm{second}\mathrm{cone}}=\frac{5}{4}\times \frac{3}{8}=\frac{15}{32} \end{gathered}$$