Question

A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is

Options

• 2h

• \[\frac{3h}{2}\]

• \[\frac{h}{2}\]

• \[\frac{2h}{3}\]

Answer 1

Disclaimer: In the the question, the statement given is incorrect. Instead of total height of solid being equal to 3 times the volume 
of cone, the volume of the total solid should be equal to 3 times the volume of the cone.

Let x be the height of cylinder.

Since, volume of the total solid should be equal to 3 times the volume of the cone, 
So,

\[\frac{1}{3} \pi r^2 h + \pi r^2 x = 3\left( \frac{1}{3} \pi r^2 h \right)\]

\[ \Rightarrow \frac{1}{3} \pi r^2 h - \pi r^2 h + \pi r^2 x = 0\]

\[ \Rightarrow \pi r^2 x = \frac{2}{3} \pi r^2 h\]

\[ \Rightarrow x = \frac{2}{3}h\]

Hence, the height of cylindrical part `= (2h)/3`