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. $2.25\pi$
• C. $2.35$
• D. Can't be determined

Answer 1

The correct option is B $2.25\pi$

Cube here will be inscribed in a cone as a square is in isosceles triangle.
Let the height of the cone be $h$
Radius=$\sqrt{2}h$
Volume of cone=$\frac{1}{3}\pi r^{2}h$
=$\frac{2\sqrt{2}}{3}\pi h^3$
Let the side of the cube be x,the top of the cone above it has the sign $(h-x)$ and radius $\frac{x}{2}$
Using properties of similar triangle $\frac{\frac{x}{2}}{h-x}=\frac{\sqrt{2}h}{h}$
$=\sqrt{2}x$
$=\frac{2\sqrt{2}h}{2\sqrt{2}+1}$
Volume of the cube=$\frac{2\sqrt{2}h}{2\sqrt{2}+1}$
Ratio of the volume of the cone to volume of the cube=$\frac{\frac{2\sqrt{2}}{3}\pi h^3}{(\frac{2\sqrt{2}h}{2\sqrt{2}+1}{)}^3}$
$=\frac{\pi (2\sqrt{2}+1{)}^3)}{24}$
$=2.35\pi$