A plant produces aluminum, facing costs of C(A) = \frac{A^2}{10}, where A is the number of aluminum units produced, and selling them on the competitive market for a price \$ 30. A farmer grows beets next to the aluminum factory, facing costs of C(B) = 5B + \frac{B^2}{100} + \frac{A^2}{20}, where B is the number of beets produced, and selling them for \$ 15 on the competitive market. Because aluminum pollution is harmful to agricultural production, the state government imposes a tax on aluminum production. What is the socially optimal tax on aluminum?

You are the manager of the packaging department in a cookie factory. (obviously, the packaging employees cannot eat the cookies that are transferred in during the period.) after your employees insert cookies into colorful packages (step 1) for display on store shelves, the packages of cookies are then boxed using cardboard cartons (step 2) for shipment to stores. each unit of product is represented by a carton of packaged cookies. the packaging department began the period with 1,000 units of cookies. during the period, 5,000 units of cookies were transferred in from the baking department and 5,500 units of cookies were transferred out to the finished goods department. the number of units of cookies in the ending inventory of the packaging department equals: