Find the angle between the normals to the surface at (1, 4, 2) and (-3, -3, 3).
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First, we express the curve in the form , where is a constant.
Thus,
Next, we find the normal vector to , which is the gradient of , that is .
Thus, .
The normal vector at (1, 4, 2) is , and the normal vector at (-3, -3, 3) is .
Finally, the angle between the two vectors and is given by
.
Therefore, the angle between the normals to the surface at (1, 4, 2) and (-3, -3, 3) is approximately .