Find the angle between two surfaces:
To find the angle between two surfaces, we need to first find the normal vectors to each surface at the given point.

Surface Equations:
1. Surface 1: x^2 + y^2 + z^2 = 9
2. Surface 2: x^2 + y^2 - z = 3

Find Normal Vectors:
1. For Surface 1:
The gradient of surface 1 is given by ∇f = (2x, 2y, 2z).
At the point (2, -1, 2), the normal vector is (∇f) = (4, -2, 4).
2. For Surface 2:
The gradient of surface 2 is given by ∇g = (2x, 2y, -1).
At the point (2, -1, 2), the normal vector is (∇g) = (4, -2, -1).

Calculate Angle:
The angle between two vectors can be calculated using the dot product formula:
θ = cos^(-1)((∇f) . (∇g) / ||∇f|| ||∇g||)
Substitute the values:
θ = cos^(-1)((4*4 + (-2)*(-2) + 4*(-1)) / sqrt(4^2 + (-2)^2 + 4^2) sqrt(4^2 + (-2)^2 + (-1)^2))
θ = cos^(-1)((16 + 4 - 4) / sqrt(16 + 4 + 16) sqrt(16 + 4 + 1))
θ = cos^(-1)(16 / sqrt(36) sqrt(21))
θ = cos^(-1)(16 / (6 * sqrt(21)))
θ ≈ cos^(-1)(0.761)
θ ≈ 40.9 degrees
Therefore, the angle between the surfaces x^2 + y^2 + z^2 = 9 and x^2 + y^2 - z = 3 at the point (2, -1, 2) is approximately 40.9 degrees.

3+4=7