Question

The volume of a sphere and cone are equal. The diameter of the sphere is equal to the diameter of the base of the cone. Find the ratio of the height of the cone to the diameter of the sphere.

Answer 1

Dear student
$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}: \\ \mathrm{Volume}\mathrm{of}\mathrm{sphere}=\frac{4}{3}{\mathrm{\pi r}}^3,\mathrm{where}\mathrm{r}\mathrm{is}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{sphere} \\ \mathrm{Voume}\mathrm{of}\mathrm{cone}=\frac{1}{3}{{\mathrm{\pi r}}_1}^2\mathrm{h},\mathrm{where}{\mathrm{r}}_1\mathrm{is}\mathrm{the}\mathrm{radius}\mathrm{and}\mathrm{h}\mathrm{is}\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{cone} \\ \mathrm{Given}:\mathrm{Volume}\mathrm{of}\mathrm{sphere}=\mathrm{Volume}\mathrm{of}\mathrm{cone} \\ \Rightarrow \frac{4}{3}{\mathrm{\pi r}}^3=\frac{1}{3}{{\mathrm{\pi r}}_1}^2\mathrm{h} \\ \Rightarrow 4{\mathrm{r}}^3={{\mathrm{r}}_1}^2\mathrm{h}...\left(1\right) \\ \mathrm{Also},\mathrm{given}\mathrm{that}:2\mathrm{r}=2{\mathrm{r}}_1 \\ \Rightarrow \mathrm{r}={\mathrm{r}}_1...\left(2\right) \\ \mathrm{To}\mathrm{find}:\frac{\mathrm{h}}{2\mathrm{r}} \\ \mathrm{Putting}\left(2\right)\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ 4{\mathrm{r}}^3={\mathrm{r}}^2\mathrm{h} \\ \mathrm{r}=\frac{\mathrm{h}}{4} \\ \mathrm{h}=4\mathrm{r}....\left(3\right) \\ \mathrm{Consider},\frac{\mathrm{h}}{2\mathrm{r}} \\ =\frac{4\mathrm{r}}{2\mathrm{r}}\left[\mathrm{using}\left(3\right)\right] \\ =2 \\ \mathrm{So},\frac{\mathrm{h}}{2\mathrm{r}}=\frac{2}{1} \end{gathered}$$
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