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?

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Solution

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, $h=2d\phantom{\rule{0ex}{0ex}}\frac{d}{h}=\frac{2}{1}\phantom{\rule{0ex}{0ex}}d:h=2:1$Hence, the ratio of diameter of the sphere to the height of the cone is $2:1$.

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