Question

a sphere and a cube have same total surface area. show that the ratio of volume of sphere to that of cube is √6:[tex] \pi

Answer 1

Let r be radius of sphere
Let a be the edge of sphere
We know,
A sphere and a cube have same total surface area.
$4 \pi r^2 = 6a^2$
$(\frac{r}{a})^2 = \frac{3}{2 \pi }$
$\frac{r}{a} = \sqrt{ \frac{3}{2 \pi } }$
$\frac{Volume of sphere}{Volume of cube} = \frac{(\frac{4}{3}) \pi r^3}{a^3} = (\frac{4 \pi }{3}) ( \frac{r}{a})^3$
$= (\frac{4 \pi }{3}) ( \sqrt{ (\frac{3}{2 \pi })^3 }$
$= (\frac{4 \pi }{3}) ( \frac{3}{2 \pi }) [ \sqrt{ (\frac{3}{2 \pi }) } ]$
$= 2 \sqrt{ \frac{3}{2 \pi } }$
$= \sqrt{ \frac{4 * 3}{2 \pi } }$
$= \sqrt{ \frac{6}{ \pi } }$
∴ Volume of sphere : volume of cube 
$= \sqrt{6} : \sqrt{ \pi }$