Question

A cone and a sphere have equal radii and equal volume. What is the ratio of the diameter of the sphere to the height of a cone?

Answer 1

It is given that, the cone and sphere have the same radii.

We know that,

$\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cone}=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$

$\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{sphere}=\frac{4}{3}{\mathrm{\pi r}}^3$

Now, the volumes of both the shapes is equal

$\begin{array}{rcl}\therefore \mathrm{Volume}\mathrm{of}\mathrm{cone}& =& \mathrm{Volume}\mathrm{of}\mathrm{sphere}\\ \frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}& =& \frac{4}{3}{\mathrm{\pi r}}^3\\ \mathrm{h}& =& 4\mathrm{r}\\ \mathrm{h}& =& 2\left(2\mathrm{r}\right)\end{array}$

Since, $\begin{array}{rcl}\because \mathrm{d}& =& 2\mathrm{r}\end{array}$

Therefore,

$$\begin{gathered} h=2d \\ \frac{d}{h}=\frac{2}{1} \\ d:h=2:1 \end{gathered}$$

Hence, the ratio of diameter of the sphere to the height of the cone is $2:1$.