Figure 12(b) shows an open rectangular box with a square base. Consider the problem of finding the values of x and h for which the volume is 32 cubic feet and the total surface area of the box is minimal. (The surface area is the sum of the areas of the five faces of the box.)

(a) Determine the objective and constraint equations.

(b) Express the quantity to be minimized as a function of x.

(c) Find the optimal values of x and h.

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