Question

If the volumes of two cones are in the ratio of 1:4 and their diameters are in the ratio of 4:5, then find the ratio of their heights.

Answer 1

$$\begin{gathered} \mathrm{Let}r\mathrm{and}R\mathrm{be}\mathrm{the}\mathrm{base}\mathrm{radii},h\mathrm{and}H\mathrm{be}\mathrm{the}\mathrm{heights},v\mathrm{and}V\mathrm{be}\mathrm{the}\mathrm{volumes}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{given}\mathrm{cones}. \\ \mathrm{We}\mathrm{have}, \\ \frac{2r}{2R}=\frac{4}{5}or\frac{r}{R}=\frac{4}{5}.....\left(\mathrm{i}\right) \\ \mathrm{and} \\ \frac{v}{V}=\frac{1}{4} \\ \Rightarrow \frac{\left({\displaystyle \frac{1}{3}}\pi r^{2}h\right)}{\left(\frac{1}{3}\pi R^{2}H\right)}=\frac{1}{4} \\ \Rightarrow \frac{{\displaystyle r^{2}h}}{R^{2}H}=\frac{1}{4} \\ \Rightarrow {\left(\frac{r}{R}\right)}^2\times \frac{h}{H}=\frac{1}{4} \\ \Rightarrow {\left(\frac{4}{5}\right)}^2\times \frac{h}{H}=\frac{1}{4}\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ \Rightarrow \left(\frac{16}{25}\right)\times \frac{h}{H}=\frac{1}{4} \\ \Rightarrow \frac{h}{H}=\frac{1\times 25}{4\times 16} \\ \Rightarrow \frac{h}{H}=\frac{25}{64} \\ \therefore h:H=25:64 \end{gathered}$$

So, the ratio of their heights is 25:64.