#1: (Section 3.3) The mountainous country of East Matrix can grow only two crops for export, coffee and cocoa. The country has 500,000 hectares of land available for crops. Long-term contracts require that at least 100,000 hectares be devoted to coffee and at least 200,000 hectares to cocoa. Cocoa must processed locally, and production bottlenecks limit cocoa to 270,000 hectares. Coffee requires two workers per hectare, with cocoa requiring five. No more than 1,750,000 people are available for working with these crops. Coffee produces a profit of $220 per hectare and cocoa a profit of $550 per hectare. a) Set-up the chart that organizes the above information. b) From the chart, state the inequalities that describe the information. c) Graph the inequalities. Be sure to choose an appropriate scale on each axis so that you can see the feasible region well. d) From the graph, determine how many hectares the country, East Matrix, should devote to each crop in order to maximize profit. e) State this maximum profit.
#2: (Section 4.1) Set-up the following problem for solution by the Simplex method. Express the linear constraints and objective function. Then, add slack variables that convert each constraint into a linear equation. Finally, set-up the initial Simplex tableau. It is not required to solve it.
The Finite Bicycle Manufacturing Company builds racing, touring, and mountain bicycle models. The bicycles are made of both aluminum and steel. The company has available 91,800 units of steel and 42,000 units of aluminum. The racing, touring, and mountain models need 17, 27, and 34 units of steel, and 12, 21, and 15 units of aluminum, respectively. How many of each type of bicycle should be made in order to maximize profit if the company makes $8 per racing bike, $12 per touring bike, and $22 per mountain bike? What is the maximum possible profit?