Question

A cone of semi-vertical angle $\alpha$ is inscribed in a sphere of radius $2$ cm. The height of the cone is

• A. $4{\sin }^2\alpha$
• B. $4{\cos }^2\alpha$
• C. $4\sin \alpha$
• D. $4\cos \alpha$

Answer 1

The correct option is B $4{\cos }^2\alpha$

Al=OL=2(radius of the sphere)
$\angle ALM=2\alpha$(exterior angle of the triangle is the same as sum of the interior opposite angles)
$\therefore LM=OL\cos 2\alpha =2\cos 2\alpha$
height of the cone
$=OM=OL+LM=2+2\cos \alpha$
$=2(1+\cos 2\alpha )$
$=2\times 2{\cos }^2\alpha =$
$=4{\cos }^2\alpha$