Question

The height of a right circular cone of maximum volume that can be enclosed within a hollow sphere of radius \(R\) is

• A. \(R\)
• B. \(\frac{5}{4}R\)
• C. \(\frac{4}{3}R\)
• D. \(\frac{3}{2}R\)

Hint:

To maximize the volume of a cone inscribed in a sphere, use the geometry of the sphere to express the radius of the cone in terms of the height and differentiate the volume equation.

Answer 1

The Correct Option is C

Step 1: Understanding the problem.
We are given a hollow sphere of radius \(R\) and need to find the height of the cone that maximizes the volume when inscribed within the sphere. The volume of a cone is given by: \[ V = \frac{1}{3} \pi r^2 h \] where \(r\) is the radius of the base and \(h\) is the height of the cone. Step 2: Using geometry.
For a cone inscribed in a sphere, the relationship between the radius of the base \(r\) and the height \(h\) is determined by the Pythagorean theorem: \[ r^2 + \left( \frac{h}{2} \right)^2 = R^2 \] Step 3: Maximizing the volume.
Maximizing the volume requires finding the critical point of the volume equation with respect to \(h\). After differentiating and solving, the maximum volume occurs when \(h = \frac{4}{3}R\).