Question

The total surface area of a cube is numerically equal to the. surface area of a sphere then the ratio of their volume is

• A. $\frac{\pi }{6}$
• B. $\sqrt{\frac{\pi }{6}}$
• C. $\frac{\pi }{216}$
• D. $\sqrt{\frac{6}{\pi }}$

Answer 1

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

TSA of a cube $=6a^2$

SA of a sphere $=4\pi r^2$

TSA of a cube $=$ SA of a sphere

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

$3a^2=2\pi r^2$

$\sqrt{3}a=\sqrt{2\pi }r$

So, $a=\sqrt{\frac{2\pi }{3}}r$

Volume of a cube $=a^3={\left(\sqrt{\frac{2\pi }{3}}r\right)}^3={\left(\frac{2\pi }{3}\right)}^{\left(\frac{2}{3}\right)}{r}^3$

Volume of a sphere $=\frac{4}{3}\pi r^3$

Required Ratio $=\frac{{\left(\frac{2\pi }{3}\right)}^{\left(\frac{2}{3}\right)}{r}^3}{\frac{4}{3}\pi r^3}=\sqrt{\frac{\pi }{6}}$