Question

The volume of how many cylinders of half the radius of the base and one-third the height of a cone is equal to the volume of the cone?

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{R}=\mathrm{radius}\mathrm{of}\mathrm{base}\mathrm{of}\mathrm{cone} \\ \mathrm{Let}\mathrm{H}=\mathrm{height}\mathrm{of}\mathrm{cone} \\ \mathrm{Volume}\mathrm{of}\mathrm{cone}=\frac{1}{3}{\mathrm{\pi R}}^2\mathrm{H} \\ \mathrm{Let}\mathrm{r}=\mathrm{radius}\mathrm{of}\mathrm{base}\mathrm{of}\mathrm{cylinder} \\ \mathrm{Now},\mathrm{r}=\frac{\mathrm{R}}{2} \\ \mathrm{Let}\mathrm{h}=\mathrm{height}\mathrm{of}\mathrm{cylinder} \\ \mathrm{h}=\frac{\mathrm{H}}{3} \\ \mathrm{Volume}\mathrm{of}\mathrm{cylinder}={\mathrm{\pi r}}^2\mathrm{h}=\mathrm{\pi }{\left(\frac{\mathrm{R}}{2}\right)}^2\left(\frac{\mathrm{H}}{3}\right)=\frac{1}{12}{\mathrm{\pi R}}^2\mathrm{H} \\ \mathrm{Required}\mathrm{no}.\mathrm{of}\mathrm{cylinders}=\frac{\mathrm{volume}\mathrm{of}\mathrm{cone}}{\mathrm{volume}\mathrm{of}\mathrm{cylinder}}=\frac{{\displaystyle \frac{1}{3}}{\mathrm{\pi R}}^2\mathrm{H}}{{\displaystyle \frac{1}{12}}{\mathrm{\pi R}}^2\mathrm{H}}=\frac{12}{3}=4 \\ \mathbf{Hence}\mathbf{,}\mathbf{}\mathbf{no}\mathbf{.}\mathbf{}\mathbf{of}\mathbf{}\mathbf{cylinders}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4} \end{gathered}$$