Solve the following problems completely. 1. A carpenter makes bookcases in 2 sizes, large and small. It takes 4 hours to make a large bookcase and 2 hours to make a small one. The profit on a large bookecase is P350 and on a small bookcase is P200. The carpenter can spend only 32 hours per week making bookcases and must make at least 2 of the large and at least 4 of the small each week. How many small and large bookcases should the carpenter make to maximize his profit? What is his profit? 2. In order to ensure optimal health (and thus accurate test results), a lab technician needs to feed the rabbits a daily diet containing a minimum of 24 grams (g) of fat, 36 g of carbohydrates, and 4g of protein. But the rabbits should be fed no more than five ounces of food a day. Rather than order rabbit food that is custom blended, it is cheaper to order Food X and Food Y, and blend them for an optimal mix. Food X contains 8 g of fat, 12 g of carbohydrates, and 2 g of protein per ounce, and costs P200 per ounce. Food Y contains 12 g of fat, 12 g of carbohydrates, and 1 g of protein per ounce, at a cost of P300 per ounce. What is the optimal blend of Food X and Food Y? 3. Last year, a city had three school districts: North with a population of 5200 children, South with a population of 10600 children, West, with a population of 15100 children. Use Hamilton, Adams, Jefferson, Webster, and Hill-Huntington methods to apportion 50 speech therapists among the districts.4. In the following graphs, determine if there is a Euclidean path, Euclidean circuit, or neither. Name at least 1 Euclidean path or circuit, if it exists. 5. The graph below has exactly two vertices of odd degree, namely A and B. Eulerize this graph to find the most efficient path that starts and ends at the same vertex and traverses each edge at least once. 6. When installing fiber optics, some companies will install a sonet ring; a full loop of cable connecting multiple locations. This is used so that if any part of the cable is damaged it does not interrupt service, since there is a second connection to the hub. A company has 5 buildings. Costs (in thousands of doliars) to lay cables between pairs of buildings are shown below. Find the circuit that will minimize cost