Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2R)/(sqrt(3))`. Also find the maximum volume.

Explanation

To solve the problem of finding the height of the cylinder of maximum volume that can be inscribed in a sphere of radius \( R \), we will follow these steps:

### Step 1: Understanding the Geometry
Consider a cylinder inscribed in a sphere of radius \( R \). Let the height of the cylinder be \( h \) and the radius of the cylinder be \( r \). The center of the sphere is at the origin, and the cylinder is symmetric about the center.

### Step 2: Relating the Cylinder and Sphere Dimensions
Using the Pythagorean theorem, we can relate the radius of the sphere, the height of the cylinder, and the radius of the cylinder. The relationship can be expressed as:
\[
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