Problem 2 You are a marketing analyst for a company that produces and sells two styles of umbrellas. Style x sells for $20 each and style 3; sells for $25 each. The company has determined that if 3: thousand of the first style and y thousand of the second style are produced, then the total cost in thousands of dollars is given by the function 3 C(x,y) = 3x2 3xy + Eyz + 32x 29y + 70 Your manager would like you to determine how many of each style of umbrella should the company produce and sell in order to maximize prot? Problem 5 International Airports across the world are combining both workers and machines to process travelers entering various countries. Salt Lake International is investigating whether they should go this route. They currently employ workers only to process travelers entering the USA through Salt Lake International. On average, from 10:00-14:00 (4 hours each day), approximately 1000 travelers coming from a foreign country enter into the USA at SLC international airport. During this 4 hour time period, SLC international hires workers to process the travelers and pays each worker $100. Using customs data from other airports, you have estimated the following production function to process travelers using machines, M, and workers, L. Q = 50L0.6M0.4 where Q represents travelers (e.g., 1000 travelers currently). Each workers costs $100 and each machine costs $800. The cost function of processing travelers with machines and workers is C = 100L + 800M You have been hired to make a decision for the airport. Should they invest in new machinery at $800 per machine? If so, what is the optimal mix of workers and machines that minimizes the cost of processing 1000 travelers. Problem 6 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, Sock- um 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