A rectangular box of dimensions x, y,z has volume V(x, y, z) = xyz and surface...

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A rectangular box of dimensions x, y,z has volume V(x, y, z) = xyz and surface area xy +2xz + 2yz (its lid is absent, and x, y, z represent its base length and depth, and its height, resp.). For a fixed surface area of 108 show, using the method of Lagrange Multipliers, that the dimensions yielding maximum volume of the box are x = y = 6,z = 3. A rectangular box of dimensions x, y,z has volume V(x, y, z) = xyz and surface area xy +2xz + 2yz (its lid is absent, and x, y, z represent its base length and depth, and its height, resp.). For a fixed surface area of 108 show, using the method of Lagrange Multipliers, that the dimensions yielding maximum volume of the box are x = y = 6,z = 3.