Final answer:
The equation of a sphere with center (3, -12, 6) and radius 10 is (x - 3)^2 + (y + 12)^2 + (z - 6)^2 = 10^2, which simplifies to x^2 - 6x + y^2 + 24y + z^2 - 12z = -99.
Step-by-step explanation:
To find the equation of a sphere with a given center (3, -12, 6) and radius 10, we use the standard equation of a sphere which is (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) are the coordinates of the center of the sphere and r is the radius.
Substituting the given values, we get:
(x - 3)² + (y + 12)² + (z - 6)² = 10²
Which simplifies to:
x² - 6x + y² + 24y + z² - 12z = -99
This is the equation of the sphere we are looking for.