Question

The area of whole surface of a cube and a sphere are equal. Find the ratio of their volumes.​

Answer 1

$\bigstar\huge\bf\underline{\underline{Answer:-}}$

$\sf Radius \ of \ a \ sphere \ = \ r \ unit$

$\sf Edge \ of \ cube \ = \ a \ unit$

$\sf Surface \ area \ of \ sphere \ = \ 4\pi r^{2}$

$\sf Surface \ area \ of \ cube \ = \ 6a^{2}$

$\sf 4\pi r^{2} \ = \ 6a^{2}$

$\sf\longmapsto \frac{a^{2}}{r^{2}} \ = \ \frac{4\pi}{6}$

$\sf\longmapsto \frac{a}{r} \ = \ \sqrt{\frac{2\pi}{3}}$

$\sf Ratio \ of \ their \ volumes$

$\sf\hookrightarrow \frac{a^{3}}{\frac{4}{3}\pi r^{3}}$

$\sf\hookrightarrow \frac{3a^{3}}{4\pi r^{2}}$

$\sf\hookrightarrow \frac{3}{4\pi}(\sqrt{\frac{2\pi}{3}})^{3}$

$\sf\hookrightarrow \frac{3}{4\pi}\times\frac{2\pi}{3}\times\frac{\sqrt{2\pi}}{\sqrt{3}}$

$\sf\hookrightarrow \frac{1}{2}\times\frac{\sqrt{2\pi}}{\sqrt{3}}$

$\Longrightarrow\boxed{\boxed{\bf \frac{\sqrt{\pi}}{\sqrt{6}}}}$

$\sf\therefore\underline{\underline{Volume \ of \ cube \ : \ Volume \ of \ sphere \ = \sqrt{\pi}:\sqrt{6}}}$