Find the height h, radius r, and volume of a right circular cylinder with maximum volume that is inscribed in a sphere of radius R.

Verified step by step guidance

1. Start by visualizing the problem: draw a sphere of radius R and inscribe a right circular cylinder within it. Identify the relationships between the cylinder's dimensions (height h and radius r) and the sphere's radius R.
2. Use the Pythagorean theorem to relate the radius of the sphere, the radius of the cylinder, and half the height of the cylinder. The relationship can be expressed as: R^2 = r^2 + (h/2)^2.
3. Set up the volume V of the cylinder as a function of its dimensions: V = πr^2h. To maximize the volume, express h in terms of r using the relationship from step 2.
4. Substitute the expression for h into the volume formula to get V as a function of r only: V(r) = πr^2(2√(R^2 - r^2)).
5. Differentiate the volume function V(r) with respect to r, set the derivative equal to zero to find critical points, and analyze these points to determine the maximum volume.