Practice writing systems of linear inequalities, finding vertices of feasible regions, and using objective functions to solve optimization problems.
Read each problem carefully. Define variables when needed, show your work, and state the final answer in context.
Model constraints, find feasible regions, and optimize objective functions
Math - Grade 9-12
- 1
For the constraints x >= 0, y >= 0, x + y <= 10, and 2x + y <= 14, list the vertices of the feasible region.
- 2
Use the feasible region from problem 1 to maximize P = 3x + 2y.
- 3
A workshop makes chairs and tables. Let x be the number of chairs and y be the number of tables. Each chair takes 2 hours and 3 boards. Each table takes 4 hours and 8 boards. The workshop has at most 40 hours and 72 boards. A chair earns $15 profit and a table earns $40 profit. Write the constraints and the objective function.
- 4
Solve the workshop problem from problem 3. Find the maximum profit and the number of chairs and tables that give it.
- 5
Decide whether the point (3, 5) is feasible for the constraints x + 2y <= 12, 3x + y <= 14, x >= 0, and y >= 0. Show your substitution.
- 6
Minimize C = 5x + 8y subject to x + y >= 12, x >= 4, and y >= 2.
- 7
A feasible region is in the first quadrant, below the line y = 12 - 2x, and below the line y = 6 - 0.5x. Write the system of inequalities.
- 8
For the system x >= 0, y >= 0, y <= 12 - 2x, and y <= 6 - 0.5x, find the vertices of the feasible region.
- 9
Use the feasible region from problem 8 to maximize R = 40x + 30y.
- 10
Explain why linear programming problems usually test only the vertices of a bounded feasible region when finding a maximum or minimum.
- 11
A meal plan uses food A and food B. Let x be servings of food A and y be servings of food B. Food A has 2 units of protein and 1 unit of fiber per serving. Food B has 1 unit of protein and 2 units of fiber per serving. The plan needs at least 10 units of protein and at least 8 units of fiber. Food A costs $3 per serving and food B costs $2 per serving. Write the linear programming model and find the minimum cost.
- 12
A linear programming solution for a transportation plan gives x = 2.5 buses and y = 4 vans. Explain why this may not be a usable final answer and what should be done next.