Question

a sphere and a cube have the same surface show that ratio of the volume of the sphere to that of cube is :
$\sqrt{6} : \sqrt{\pi}$

Answer 1

a sphere and a cube have same surface area

total surface area of sphere=total surface area of cube
$4\pi {r}^{2} = 6 {l}^{2}$

$\frac{ {r}^{2} }{ { l}^{2} } = \frac{6}{4\pi}$

$\frac{r}{l} = \frac{ \sqrt{6} }{2 \sqrt{\pi} } .......................(1)$


$\frac{volume \: of \: sphere}{volume \: of \: cube} = \frac{ \frac{4}{3} \times \pi {r}^{3} }{ {l}^{3} }$


$= \frac{4}{3} \pi \times {( \frac{r}{l}) }^{3}$

$= \frac{4}{3} \pi \times ({ \frac{ \sqrt{6} }{2 \sqrt{\pi} }) }^{3}$

$= \frac{4}{3} \pi \times \frac{ \sqrt{6 } \times \sqrt{6} \times \sqrt{6} }{8 \times \sqrt{\pi} \times \sqrt{\pi} \times \sqrt{\pi} }$

$= \frac{4}{3}\pi \times \frac{6 \times \sqrt{6} }{8 \times \pi \times \sqrt{\pi} }$

$= \frac{2 \sqrt{6} }{2 \sqrt{\pi} }$

$= \frac{ \sqrt{6} }{ \sqrt{\pi} }$
hence, proved

hope this helps