In the vat design problem, suppose that 48 square meters of sheet metal is available. Show that if the shadow price of sheet metal is 1, designing a vat to maximize the net profit function produces a vat of the optimal shape and exactly exhausts the available metal. Example 5.27 gives the fundamental economic insight of the Lagrangean method. We impute a price to each constraint and maximize the net benefit function

consisting of the objective function minus the imputed value of the constraints. The first-order conditions for unconstrained maximization of the net benefit (42) define the optimal value of x in terms of the shadow prices λ. Together with the constraints, this enables us to solve for both the optimal x and the optimal shadow prices.

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L(x, λ) = f(x)-〉-Ag(x) (42)