Question

The radius and slant height of a cone are in the ratio 1:3. The volume of cone is 452.16 root 2 cm³. Find the height of the cone. (Take pie = 3.14)

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{base}\mathrm{of}\mathrm{cone}=\mathrm{r} \\ \mathrm{Let}\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{cone}=\mathrm{h} \\ \mathrm{Let}\mathrm{the}\mathrm{slant}\mathrm{height}\mathrm{of}\mathrm{cone}=\mathrm{l} \\ \mathrm{Now}, \\ \frac{\mathrm{r}}{\mathrm{l}}=\frac{1}{3} \\ \Rightarrow \mathrm{l}=3\mathrm{r}.....\left(1\right) \\ \mathrm{Now},{\mathrm{h}}^2={\mathrm{l}}^2-{\mathrm{r}}^2={\left(3\mathrm{r}\right)}^2-{\mathrm{r}}^2=9{\mathrm{r}}^2-{\mathrm{r}}^2=8{\mathrm{r}}^2 \\ \Rightarrow \mathrm{h}=2\sqrt{2}\mathrm{r} \\ \mathrm{Now},\mathrm{volume}\mathrm{of}\mathrm{cone}=452.16\sqrt{2}{\mathrm{cm}}^3 \\ \Rightarrow \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}=452.16\sqrt{2} \\ \Rightarrow \frac{1}{3}\times 3.14\times {\mathrm{r}}^2\times 2\sqrt{2}\mathrm{r}=452.16\sqrt{2} \\ \Rightarrow {\mathrm{r}}^3=\frac{452.16\sqrt{2}\times 3}{3.14\times 2} \\ \Rightarrow {\mathrm{r}}^3=216 \\ \Rightarrow \mathrm{r}=6\mathrm{cm} \\ \mathrm{Now},\mathrm{height},\mathrm{h}=2\sqrt{2}\mathrm{r}=2\sqrt{2}\times 6=12\sqrt{2}\mathrm{cm}=12\times 1.414=16.968\mathrm{cm} \end{gathered}$$