If two cylinders of equal volumes have their heights in the ratio $2 : 3$, then the ratio of their radii is

### Concept & Formula When two cylinders have the same volume, the inverse relationship between their heights and the square of their radii can be used to find the ratio. $$V = \pi r^2 h$$ ### Step-by-Step Solution 1. **Equate the volumes:** - $V_1 = V_2$ - $\pi r_1^2 h_1 = \pi r_2^2 h_2$ 2. **Rearrange to find the ratio of radii squared:** - $\frac{r_1^2}{r_2^2} = \frac{h_2}{h_1}$ 3. **Substitute the given height ratio:** - We are given $h_1 : h_2 = 2 : 3$, so $\frac{h_2}{h_1} = \frac{3}{2}$. - $\frac{r_1^2}{r_2^2} = \frac{3}{2}$ 4. **Take the square root of both sides:** - $\frac{r_1}{r_2} = \sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}$ - The ratio is $\sqrt{3} : \sqrt{2}$. ### Exam Strategy & Shortcut If volumes are equal, $r \propto \frac{1}{\sqrt{h}}$. Therefore, the ratio of the radii is exactly the square root of the inverse ratio of their heights. ### Common Pitfall Taking the direct square root of the height ratio ($\sqrt{2} : \sqrt{3}$) without inverting it first. Always ensure you cross-multiply correctly. ### Final Answer Therefore, the correct answer is **$\sqrt{3} : \sqrt{2}$**.