Final answer:
To find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 9y + 4z = 27, we can use the method of Lagrange multipliers. The objective function is the volume of the box, and the constraint is the equation of the plane. By setting up and solving the Lagrangian function, we can find the maximum volume of the box.
Step-by-step explanation:
To find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 9y + 4z = 27, we can use the method of Lagrange multipliers. The objective function is the volume of the box, which is given by V = (2Lx)(2Ly)(Lz) = 8LxLyLz. The constraint is the equation of the plane: x + 9y + 4z = 27. We can set up the Lagrangian function as follows:
L(Lx, Ly, Lz, λ) = 8LxLyLz + λ(x + 9y + 4z - 27)
We can then find the partial derivatives with respect to all variables and set them equal to zero:
∂L/∂Lx = 8LyLz + λ = 0
∂L/∂Ly = 8LxLz + 9λ = 0
∂L/∂Lz = 8LxLy + 4λ = 0
∂L/∂λ = x + 9y + 4z - 27 = 0
Solving this system of equations will give us the values of Lx, Ly, Lz, and λ. Substituting these values back into the volume equation will give us the maximum volume of the rectangular box.