1. A cardboard container is being designed. The container will be arectangular shape, divided into 12 smaller rectangularcompartments. The bottom of the box must be a fixed area A, andstrips of cardboard will be needed to form the walls and thedividers inside the box. In order to minimize costs, the containermust be designed to minimize the length of cardboard used to formthe edges and dividers.

   1. (a) Assume the box will be divided into a 12 by 1 grid.Determine the minimum length of cardboard needed to form the sidesand dividers of the box.

   2. (b) Assume the box will be divided into a 6 by 2 grid. Determinethe minimum length of cardboard needed to form the sides anddividers of the box.

   3. (c) Assume the box will be divided into a 4 by 3 grid. Determinethe minimum length of cardboard needed to form the sides anddividers of the box.

   4. (d) Do the dimensions of the grid affect the minimum amount ofcardboard needed? If so, which shape is most efficient?

2. The demand for a certain product at a particular retail locationis estimated to be 64,000 items for the coming year. The product isstored cheaply in a warehouse, but it costs $0.80 per item per yearto store them on location at the retailer. The retailer wishes tominimize costs by keeping fewer items in stock on location. Theretailer must choose how many shipments per year to order from thewarehouse. Each shipment will cost $100, regardless of the size ofthe shipment. Determine the number of shipments that will minimizetotal costs for the year. You may assume that the items are sold ata steady rate, and that each new shipment arrives exactly when theprevious one runs out. (Hint: Storage costs can be determined bythe cost per item per year, the storage time per shipment, and theaverage number of items in stock over this time period. The averagenumber will be half the shipment size.)