Question

1. A cardboard container is being designed. The container will be a rectangular shape, divided into 12 smaller rectangular compartments. The bottom of the box must be a fixed area A, and strips of cardboard will be needed to form the walls and the dividers inside the box. In order to minimize costs, the container must be designed to minimize the length of cardboard used to form the 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 sides and dividers of the box.

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

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

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

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

Answer 1

The solution is given below. I have used calculus, specifically differentiation to find the answer. We differentiate the expression for length and equate to 0 to find the minima.

d. From the above we see that the length is least in the case of a 4x3 grid i.e that shape is the most efficient.