First, identify the type of problem you are being asked to solve. Is it a constrained optimization problem requiring a Lagrangian setup? Would solving the problem requiring using a definite integral? Is the problem a straightforward optimization problem like maximizing profits or revenues, or minimizing costs? I need as much detail as possible when solving the problem. PLEASE SHOW YOUR MATH CALCULATIONS. please do not use ChatGPT.

You have just been hired to manage inventory for a sock company, \"Sock-um.\" The company seems to be running out of inventory space. On day one of your job, you have been asked to determine average daily inventory. Every Monday, Sock-um receives a shipment of 500 cases of socks. Sock-um sells socks to retail outlets along the East Coast at a rate of 90 cases per day during each of the 5 business days (Monday through Friday). In your memo, please provide a model that represents daily inventory and provide your answer of average daily inventory. In your worksheets, demonstrate your knowledge of integrals and show how you can double check your answer through manual calculations. Sock-um has an annual contractual obligation with its manufacturer to receive 500 cases weekly, and breaking the contract would cost them \\( \\$ 8000 \\). Once inventory has reached 500 cases, Sockum will run out of space. Currently, additional shelf space for 500 cases is \\( \\$ 1000 \\). A pair of socks costs \\( \\$ 1 \\) to manufacturer, and each pair sells for \\( \\$ 4 \\). There are 100 pair of socks in a case of socks. Sock-um has the possibility of expanding its reach to midwestern states by providing up to 20 cases per day. To do this, they will incur a one-time cost of \\( \\$ 6000 \\). Provide a marketing recommendation to Sock-um regarding the inventory constraints, contractual obligations, and sale opportunities in the Midwest