A solid sphere is perfectly inscribed inside a cube, and a right circular cylinder is placed inside the cube such that it touches all the faces of the cube. What is the total surface area of the cylinder? Statements I:  The volume of the cube is 343 cubic
cm. Statements II:  The height of the cylinder is equal to the diameter of the sphere.

Statement I: If the volume of the cube is 343 cubic cm, we can determine the side length of the cube (since side³ = volume). From the side length, we can calculate the radius of the sphere and the dimensions of the cylinder. However, this information alone does not directly give us the surface area of the cylinder without additional information about its height. Statement II: Knowing that the height of the cylinder is equal to the diameter of the sphere helps us determine the relationship between the dimensions of the cylinder and the cube. Combined with the radius (which is half the side length), this statement gives us enough information to calculate the surface area of the cylinder. Combining both statements: Together, we can determine both the radius and the height of the cylinder with the given conditions, which is sufficient to calculate the surface area. Answer: C. Both statements together are sufficient, but neither statement alone is sufficient.