EXAMPLE 1 A rectangular box without a lid is to be made from 48 m2 of cardboard. Find the maximum volume of such a box. SOLUTION We let x, y, and z to be the length, width, and height, respectively, of the box in meters. Then we wish to maximize V = xyz subject to the constraint g(x, y, z) = 2xz + 2yz + xy = 48. Using the method of Lagrange multipliers, we look for the values of x, y, z, and λ such that ∇V = λ∇g and g(x, y, z) = 48. This gives the equations vx = λgx vy = λgy vz = λgz 2xz + 2yz + xy = 48