Question

If a cube of maximum possible volume is cut off from a solid sphere of diameter d, then the volume of the remaining (waste) material of the sphere would be equal to:

• A. $\frac{d^3}{3}(\pi -\frac{d}{2})$
  
• B. $\frac{d^3}{3}(\frac{\pi }{2}-\frac{1}{\sqrt{3}})$
  
• C. $\frac{d^2}{4}(\sqrt{2}-\pi )$
• D. None of these

Answer 1

Answer: (B)

$\frac{d^3}{3}(\frac{\pi }{2}-\frac{1}{\sqrt{3}})$


The diagonal of cube will be equal to the diameter of sphere.
$\therefore$ Volume of sphere $=\frac{4}{3}\pi (\frac{d}{2}{)}^3=\frac{\pi d^3}{6}$
and each side of cube = $a=\frac{d}{\sqrt{3}}$
$\therefore$ Volume of cube $=a^3=\frac{d^3}{3\sqrt{3}}$
$\therefore$ Remianing volume $=\frac{\pi d^3}{6}-\frac{d^3}{3\sqrt{3}}=\frac{d^3}{3}(\frac{\pi }{2}-\frac{1}{\sqrt{3}})$