Question

Choose the correct answer of the following question:

If the radius of the base of a right circular cylinder is halved, keeping the height the same, then the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is

(a) 1 : 2 (b) 2 : 1 (c) 1 : 4 (d) 4 : 1 [CBSE 2012]

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{base}\mathrm{radius}\mathrm{and}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{original}\mathrm{cylinder}\mathrm{be}r\mathrm{and}h,\mathrm{respectively}. \\ \mathrm{Also}, \\ \mathrm{The}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{new}\mathrm{cylinder},R=\frac{r}{2}\mathrm{and}\mathrm{its}\mathrm{height},H=h. \\ \mathrm{Now}, \\ \mathrm{The}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{new}\mathrm{cylinder}\mathrm{to}\mathrm{that}\mathrm{of}\mathrm{the}\mathrm{original}\mathrm{cylinder}=\frac{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{new}\mathrm{cylinder}}{\mathrm{Voilume}\mathrm{of}\mathrm{the}\mathrm{original}\mathrm{cylinder}} \\ =\frac{\mathrm{\pi }{r}^{2}h}{\mathrm{\pi }{R}^{2}H} \\ =\frac{\mathrm{\pi }{r}^{2}h}{\mathrm{\pi }{\left({\displaystyle \frac{r}{2}}\right)}^{2}h} \\ =\frac{4\mathrm{\pi }{r}^{2}h}{\mathrm{\pi }{r}^{2}h} \\ =\frac{4}{1} \\ =4:1 \end{gathered}$$

Hence, the correct answer is option (d).