Final answer:
The largest sphere contained in the cube determined by the planes x=2 and x=6 has a radius of 2 units and its center is at (4, 4, 4). The equation of this sphere is (x-4)² + (y-4)² + (z-4)² = 4.
Step-by-step explanation:
To find an equation of the largest sphere contained in the cube determined by the planes x=2 and x=6, we need to determine the sphere's center and radius. The cube's edge length is the distance between the two planes, which is 6 - 2 = 4 units. Since the largest sphere would touch all sides of the cube, its diameter equals the cube's edge length, making the radius of the sphere half the edge length, radius = 2 units.
The center of the sphere will be at the center of the cube. Given the cube's bounds along the x-axis are x=2 and x=6, the center's x-coordinate is the midpoint, (2+6)/2 = 4. The sphere's center will have the same y and z coordinates since the cube's sides are equivalent, therefore the center is at (4, 4, 4).
The equation of a sphere with center (h, k, l) and radius r is given by (x-h)² + (y-k)² + (z-l)² = r². Substituting our values in, we get the equation of the largest sphere contained in the cube: (x-4)² + (y-4)² + (z-4)² = 4.