Question

Two right circular cylinders of equal volumes have their heights in the ratio 1 : 2. What is the ratio of their radii?

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{radii}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{cylinders}\mathrm{be}r\mathrm{and}R;\mathrm{and}\mathrm{the}\mathrm{heights}\mathrm{be}h\mathrm{and}H. \\ \mathrm{We}\mathrm{have}, \\ \frac{h}{H}=\frac{1}{2}.....\left(\mathrm{i}\right) \\ \mathrm{Now}, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{cylinder}=\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{second}\mathrm{sphere} \\ \Rightarrow \mathrm{\pi }{r}^{2}h=\mathrm{\pi }{R}^{2}H \\ \Rightarrow \frac{h}{H}=\frac{R^2}{r^2} \\ \Rightarrow \frac{1}{2}=\frac{R^2}{r^2} \\ \Rightarrow \frac{r^2}{R^2}=\frac{1}{2} \\ \Rightarrow {\left(\frac{r}{R}\right)}^2=\frac{2}{1} \\ \Rightarrow \frac{r}{R}=\frac{\sqrt{2}}{1} \\ \therefore r:R=\sqrt{2}:1 \end{gathered}$$

So, the ratio of their radii is $\sqrt{2}$ : 1.