A company produces open-top boxes with square bases for metal containers. The base of the boxes is made of different materials than the sides. The box is assembled by riveting a bracket at each of the eight corners. The total cost of producing a box is the sum of the cost of the materials for the box and the labor costs associated with affixing each bracket. As a consultant of the company, you need to devise a formula for the total cost of producing each box and determine the dimensions that allow a box of specified volume to be produced at minimum cost. Use the following notation to solve this problem.

Volume of the box = V

Height of the box = h

Length of sides of each base = x

Cost of the material for the base = A per square unit

Cost of the material for the sides = B per square unit

Cost of each bracket = C

1. Write an expression for the company's total cost in terms of these quantities.

2. Find a formula for each dimension of the box so that the total cost is a minimum.

3. The company needs to produce boxes of 48 cubic feet with a base material costing the company $12 per square foot and side material costing $ 8 per square foot. Each bracket cost $5, and the associated labor cost is $1 per bracket. Use your formula to find the dimension of the box that can be produced at a minimum cost. What is this cost? The company uses approximately 100,000 brackets per year, and the purchase price of each is $5. It buys the same number brackets ( say, n) each time it places an order with supplier, and it costs $60 to process each order. The company also has additional costs associated with storing, insuring, and financing its inventory of brackets. These carrying costs amount to 15% of the average value of the inventory annually. The brackets are used steadily and deliveries are made just as inventory reaches zero so that inventory fluctuates between zero and n brackets.

We have the following given values:

V = 48 cubic feet

A = 12 dollars per square foot

B = 8 dollars per square foot

C = 5 dollars per bracket

Labor cost for each bracket = 1 dollar

Yearly number of brackets = 100,000

Cost of purchase for each bracket = 5 dollar

Order Cost = 60 dollars

Carrying cost = 15% of average inventory annually

4. If the total annual cost associated with the bracket supply is the sum of the annual purchasing cost and the annual carrying costs, what order side n would minimize the total cost?

5. In the general case of the bracket-ordering problem, the order size n that minimizes the total cost of the bracket supply is called the economic order quantity, or EOQ. Use the following notations to determine the general formula for the EOQ. Fixed cost per order = F; Unit cost = C; Quantity purchased per year = P; Carrying cost (as a decimal rate) = r

(Please do not answer if you are unsure and give a full explanation if you do answer)