Final answer:
To find the point on the paraboloid where the tangent plane is parallel to the given plane, we can use the gradient vector and normal vector. Setting the normal vectors equal to each other, we can solve for x, y, and z to find the desired point.
Step-by-step explanation:
To find the point on the paraboloid where the tangent plane is parallel to the plane 3x + 2y + 5z = 8, we can use the gradient vector and normal vector of the plane. The gradient vector of the paraboloid is (2x, 0, 2z) and the normal vector of the plane is (3, 2, 5). Since the tangent plane is parallel to the plane, the normal vector of the tangent plane will be equal to the normal vector of the plane. Setting the two normal vectors equal to each other, we get the equations 2x = 3, 0 = 2, and 2z = 5. Solving these equations, we find x = 3/2, y can be any value because of the equation 0 = 2, and z = 5/2.