Question

A right circular cone is inscribed in a sphere of radius 3 units. If the volume of the cone is maximum, then the semi-vertical angle of the cone is:

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{6} \)
• C. \( \tan^{-1} (\sqrt{2}) \)
• D. \( \tan^{-1} \left(\frac{1}{\sqrt{2}}\right) \)

Hint:

The maximum volume condition for a cone inscribed in a sphere is derived using calculus. - The optimal semi-vertical angle is found using trigonometric identities.

Answer 1

The Correct Option is D

Step 1: Use the relation for the maximum volume
A cone inscribed in a sphere has its maximum volume when its semi-vertical angle \( \theta \) satisfies: \[ \tan \theta = \frac{1}{\sqrt{2}}. \] Step 2: Find \( \theta \)
Taking inverse tangent, \[ \theta = \tan^{-1} \left(\frac{1}{\sqrt{2}}\right). \] Thus, the correct answer is \( \boxed{\tan^{-1} \left(\frac{1}{\sqrt{2}}\right)} \).