Question

The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their volumes and the ratio of their curved surfaces.

Answer 1

Let there be two cylinder whose volumes are V₁ and V₂ and curved
surfaces are S₁and S₂. Let their radii be 2r and 3r and their heights be 5h and 3h.
$$\begin{gathered} \mathrm{Volume}\mathrm{of}\text{a}\mathrm{cylinder}=\mathrm{\pi }{r}^{2}h \\ So,V_1=\mathrm{\pi }{\left(2r\right)}^2\times \left(5h\right) \\ {V}_2=\mathrm{\pi }{\left(3r\right)}^2\times 3h \\ \text{R}\mathrm{atio}\text{of their volumes}\mathrm{is}\frac{V_1}{V_2}=\frac{\mathrm{\pi }{\left(2r\right)}^2\times \left(5h\right)}{\mathrm{\pi }{\left(3r\right)}^2\times 3h} \\ =\frac{20r^{2}h}{27r^{2}h} \\ =\frac{20}{27} \end{gathered}$$

$$\begin{gathered} \mathrm{Curved}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{cylinder}=2\mathrm{\pi }rh \\ So,S_1=2\mathrm{\pi }\left(2r\right)\times \left(5h\right) \\ {S}_2=2\mathrm{\pi }\left(3r\right)\times 3h \\ \text{R}\mathrm{atio}\text{of their curved surface areas is}\frac{S_1}{S_2}=\frac{2\mathrm{\pi }\left(2r\right)\times \left(5h\right)}{2\mathrm{\pi }\left(3r\right)\times 3h} \\ =\frac{20rh}{18rh} \\ =\frac{10}{9} \end{gathered}$$