Final answer:
To find the equation of the tangent plane to the surface x + y + z = 10 at (3, 5, 2), we determine that the gradient of the surface (1, 1, 1) is the normal vector to the tangent plane at the point and form the equation (1)(x - 3) + (1)(y - 5) + (1)(z - 2) = 0, which simplifies to x + y + z = 10.
Step-by-step explanation:
The question asks us to find an equation of the tangent plane to the given surface x + y + z = 10 at the point (3, 5, 2). To find the tangent plane at a particular point on the surface, we need the gradient of the surface at that point. The gradient of the surface given by the equation x + y + z = 10 is the normal vector to the surface at any point, which is (1, 1, 1) since these are the coefficients of x, y, and z in the equation. A tangent plane can be expressed by the equation A(x - x0) + B(y - y0) + C(z - z0) = 0, where (A, B, C) is the normal vector to the plane and (x0, y0, z0) is the point of tangency. By substituting the normal vector and the given point, we get the tangent plane equation: (1)(x - 3) + (1)(y - 5) + (1)(z - 2) = 0 which simplifies to x + y + z = 10, the same as the given equation of the surface, which makes sense because the surface is a plane itself and its tangent plane at any point would be the plane itself.