Final answer:
The equation of the sphere centered at (8, 2, 2) with radius 9 is (x - 8)² + (y - 2)² + (z - 2)² = 81. Its intersection with the plane z = 10 is a circle given by the equation (x - 8)² + (y - 2)² = 49.
Step-by-step explanation:
Finding the Equation of a Sphere
To find the equation of a sphere centered at (8, 2, 2) with a radius of 9, we use the standard equation of a sphere (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) is the center and r is the radius. Plugging in the values, we get (x - 8)² + (y - 2)² + (z - 2)² = 81.
Intersection with a Plane
The intersection of this sphere with the plane z = 10 can be found by substituting z = 10 into the sphere's equation: (x - 8)² + (y - 2)² + (10 - 2)² = 81. Simplifying, we get (x - 8)² + (y - 2)² = 49, which describes a circle (the intersection) with a center at (8, 2) and a radius of 7 in the xy-plane.