center of mass = (0, 0, 96/13) exactly (0, 0, 7.3846) approximately.
The object described is a spherical cap for a sphere with a radius of 10. Since the sphere is centered at the origin, the center of mass will have X and Y coordinates of 0 and we only need to find the Z coordinate. The formula for the geometric centroid of a spherical cap is:
z = 3(2R - h)^2 / 4(3R - h)
where
z = distance from the center of the sphere
R = radius of sphere
h = distance from base of spherical cap to top of spherical cap
And for a spherical cap of uniform density, the geometric centroid is also known as the center of mass.
Since the sphere has a radius of 10 and is cut by the plane z=6, the value h will be 10-6 = 4. So substitute the known values into the formula:
z = 3(2R - h)^2 / 4(3R - h)
z = 3(2*10 - 4)^2 / 4(3*10 - 4)
z = 3(20 - 4)^2 / 4(30 - 4)
z = 3(16)^2 / 4(26)
z = 3(256) / 104
z = 768/104
z = 96/13
z ~= 7.384615385
