Question

The radius of a cone is $\sqrt{2}$ times the height of the cone. A cube of maximum possible volume is cut from the same cone. What is the ratio of the volume of the cone to the volume of the cube?

• A. $3.18\pi$
• B. $2.35\pi$
• C. $2.35$
• D. Can't be determined

Answer 1

The correct option is B $2.35\pi$

Let height of cone be '$h$'. Then its radius $r=\sqrt{2}h$

Volume of cone $=\frac{1}{3}\pi r^{2}h=\frac{2\sqrt{2}\pi h^3}{3}$

Let side of cube be $x$, then top of cone has the size $(h-x)$ and radius $\frac{x}{2}$ using similar triangle property.

$\frac{\frac{x}{2}}{h-x}=\frac{\sqrt{2}h}{h}\Rightarrow x=\frac{2\sqrt{2}h}{2\sqrt{2}+1}$

Volume of cube $={\left(\frac{2\sqrt{2}h}{2\sqrt{2}+1}\right)}^3$

$\therefore$ Required ratio $=\frac{\frac{2\sqrt{2}{h}^3}{3}}{{\left(\frac{2\sqrt{2}h}{2\sqrt{2}+1}\right)}^3}=\frac{{\pi \times (2\sqrt{2}+1)}^3}{24}=2.35\pi$