Question

a sphere, right circular cylinder and a right circular cone have the same volume and height. find the ratio of their radius?

Answer 1

$$\begin{gathered} \mathrm{Let}{\mathrm{r}}_1\mathrm{be}\mathrm{the}\mathrm{rdius}\mathrm{of}\mathrm{sphere}. \\ \mathrm{Volume}\mathrm{of}\mathrm{sphere}={\mathrm{V}}_1=\frac{4}{3}{{\mathrm{\pi r}}_1}^3 \\ \mathrm{Let}{\mathrm{r}}_2\mathrm{be}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{base}\mathrm{of}\mathrm{cylinder}\mathrm{and}\mathrm{h}\mathrm{be}\mathrm{its}\mathrm{height}. \\ \mathrm{Now},\mathrm{h}=2{\mathrm{r}}_1 \\ \mathrm{So},\mathrm{volume}\mathrm{of}\mathrm{cylinder},{\mathrm{V}}_2={{\mathrm{\pi r}}_2}^2{\mathrm{h}}_2 \\ \Rightarrow {\mathrm{V}}_2={{\mathrm{\pi r}}_2}^2\left(2{\mathrm{r}}_1\right)=2{\mathrm{\pi r}}_1{{\mathrm{r}}_2}^2 \\ \mathrm{Let}{\mathrm{r}}_3\mathrm{be}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{base}\mathrm{of}\mathrm{cone}\mathrm{and}\mathrm{h}\mathrm{be}\mathrm{its}\mathrm{height}. \\ \mathrm{Now},\mathrm{h}=2{\mathrm{r}}_1 \\ \mathrm{volume}\mathrm{of}\mathrm{cone},{\mathrm{V}}_3=\frac{1}{3}\mathrm{\pi }{\left({\mathrm{r}}_3\right)}^2\mathrm{h}=\frac{1}{3}{{\mathrm{\pi r}}_3}^2\left(2{\mathrm{r}}_1\right)=\frac{2}{3}{\mathrm{\pi r}}_1{{\mathrm{r}}_3}^2 \\ \mathrm{Now},{\mathrm{V}}_1={\mathrm{V}}_2={\mathrm{V}}_3 \\ \Rightarrow \frac{4}{3}{{\mathrm{\pi r}}_1}^3=2{\mathrm{\pi r}}_1{{\mathrm{r}}_2}^2=\frac{2}{3}{\mathrm{\pi r}}_1{{\mathrm{r}}_3}^2 \\ \Rightarrow 4{{\mathrm{r}}_1}^2=6{{\mathrm{r}}_2}^2=2{{\mathrm{r}}_3}^2 \\ \Rightarrow 2{{\mathrm{r}}_1}^2=3{{\mathrm{r}}_2}^2={{\mathrm{r}}_3}^2 \\ \Rightarrow \frac{{{\mathrm{r}}_1}^2}{{{\mathrm{r}}_2}^2}=\frac{3}{2}\mathrm{and}\frac{{{\mathrm{r}}_2}^2}{{{\mathrm{r}}_3}^2}=\frac{1}{3} \\ \Rightarrow \frac{{{\mathrm{r}}_1}^2}{{{\mathrm{r}}_2}^2}=\frac{3}{2}\mathrm{and}\frac{{{\mathrm{r}}_2}^2}{{{\mathrm{r}}_3}^2}=\frac{2}{6} \\ \Rightarrow {{\mathrm{r}}_1}^2:{{\mathrm{r}}_2}^2:{{\mathrm{r}}_3}^2=3:\hspace{0.17em}2:6 \\ \Rightarrow {\mathrm{r}}_1:{\mathrm{r}}_2:\hspace{0.17em}{\mathrm{r}}_3=\sqrt{3}:\sqrt{2}:\sqrt{6} \end{gathered}$$