4. Answer the jewelry box problem below. [ 12 points] (a) A jewelry box is constructed by partitioning a box with a square base as shown in the accompanying gure. If the box is designed to have volume 1,152 cm3, express the total surface area (top, bottom, sides, and interior partitions) as a function of :1: and y. Ignore the thickness of the material. Use the method of Lagrange multipliers to identify the dimensions that will minimize its total surface area. Notice that we have said nothing about where the partitions are located. Does it matter? Suppose the jewelry box in (a) is designed so that the material in the top costs twice as much as the material in the bottom and sides and three times as much as the material in the interior partitions. If the material in the interior costs $2 per cmz, express the total cost of constructing the box as a function of x and y. Use the method of Lagrange multipliers to identify the dimensions that will minimize the total cost of constructing the box. What is the minimum total cost