Final answer:
To find the dimensions of the box that maximize volume, set up the Lagrange function and solve the system of equations to find the critical points. Substitute the values into the volume function to determine the maximum volume.
Step-by-step explanation:
To find the dimensions of the box that maximize volume, we need to maximize the volume function V = xyz while satisfying the constraint 6x + 2y + 3z = 18. We can solve this problem using the method of Lagrange multipliers. Let's set up the Lagrange function:
L(x, y, z, λ) = xyz + λ(6x + 2y + 3z - 18)
Taking partial derivatives and setting them equal to zero:
rac{{∂L}}{{∂x}} = yz + 6λ = 0
rac{{∂L}}{{∂y}} = xz + 2λ = 0
rac{{∂L}}{{∂z}} = xy + 3λ = 0
We now have a system of equations that we can solve to find the critical points. By solving this system, we can find the values of x, y, and z that maximize the volume function V = xyz. To find the maximum volume, substitute the values of x, y, and z into the volume function V = xyz.