The problem is to find the radius and height of the open right circular cylinder of largest surface area that can be inscribed in a sphere of radius a. What is the largest surface area? The open cylinder's surface area will be [tex] f(h,r) = 2 \pi r h [/tex] I am not really sure about the sphere, because I'm not really sure about the constraints that would apply. It looks like it would just be [tex] a^2 = r^2 + h^2 + r^2 = 2r^2 +h^2[/tex], but this would only be if the cylinder is centered about the origin. But I guess that since it is a sphere, and perfectly symmetrical, then trying to squeeze the cylinder in diagonally would be the same as along the origin. Right?