Question

A sphere and cube have equal surface areas. The ratio of their volumes is

• A. $\frac{\pi }{6}$
• B. $\frac{6}{\pi }$
• C. $\sqrt{\frac{6}{\pi }}$
• D. $\sqrt{\frac{\pi }{3}}$

Answer 1

The correct option is C $\sqrt{\frac{6}{\pi }}$

Surface area of sphere$=4\pi r^2$(r is radius )

Surafce area of cube$=6x^2$(a is side of cube)

Given,

Surface area of sphere=Surface area of cube

$4\pi r^2=6x^2$

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

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

$=>\frac{r}{a}=\sqrt{\frac{6}{4\pi }}$ (i)

Now.

$\frac{Volumeofsphere}{volumeofcube}=\frac{\frac{4}{3}\pi r^3}{a^3}$

$=\frac{4}{3}\pi (\frac{r}{a}{)}^3$
$=\frac{4}{3}\pi (\sqrt{\frac{6}{4\pi }}{)}^3$ (from (i)
$=\frac{4}{3}\pi \times \frac{6}{4\pi }\times \left(\sqrt{\frac{6}{4\pi }}\right)$
$=2\times \sqrt{\frac{6}{4\pi }}=\left(\sqrt{\frac{6\times 4}{4\pi }}\right)$
$=\sqrt{\frac{6}{\pi }}$