Final answer:
To find the equation for the tangent plane to the surface, we calculate the partial derivatives of the surface equation with respect to x, y, and z. Using these partial derivatives, we can construct the equation of the tangent plane using the formula (x - x0)(∂f/∂x)0 + (y - y0)(∂f/∂y)0 + (z - z0)(∂f/∂z)0 = 0. Substituting the values x0 = 3, y0 = 1, and z0 = 0, we find that the equation of the tangent plane to the surface at the point (3, 1, 0) is x + 15y - 18 = 0.
Step-by-step explanation:
To find the equation for the tangent plane to the surface, we first need to calculate the partial derivatives of the surface equation with respect to x, y, and z. The equation is given as 2 + 3 = xy5cos(z). Taking the partial derivatives and evaluating them at the point (3, 1, 0), we get:
∂f/∂x = y5cos(z), ∂f/∂y = 5xy4cos(z), ∂f/∂z = -xy5sin(z)
Using these partial derivatives, we can construct the equation of the tangent plane using the formula: (x - x0)(∂f/∂x)0 + (y - y0)(∂f/∂y)0 + (z - z0)(∂f/∂z)0 = 0. Substituting the values x0 = 3, y0 = 1, and z0 = 0, we get the equation of the tangent plane as:
(x - 3)(cos(0)) + (y - 1)(15cos(0)) + (z - 0)(0) = 0
Simplifying, we get:
x - 3 + 15y - 15 = 0
Therefore, the equation of the tangent plane to the surface at the point (3, 1, 0) is x + 15y - 18 = 0.