Show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3..

Explanation

To show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of radius \( r \) is \( \frac{4r}{3} \), we will follow these steps:

### Step 1: Understand the Geometry
We have a sphere of radius \( r \) and a cone inscribed in it. The height of the cone will be denoted as \( h \) and the radius of the base of the cone as \( R \).

### Step 2: Relate the Dimensions
From the center of the sphere to the vertex of the cone, the distance is \( r \). The relationship between the height \( h \) of the cone and the radius \( R \) of the base can be established using the Pythagorean theorem:

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