Analyzing Feasible Regions in Systems of Inequalities

A farmer has 100 acres of land. He wants to plant corn and wheat. Each acre of corn requires 2 hours of labor, and each acre of wheat requires 1 hour of labor. If he has a total of 120 hours of labor available, write a system of inequalities to represent the situation and identify the feasible region.

• x + 2y ≤ 100, 2x + 2y ≤ 120, x ≥ 0, y ≥ 0

• x + y ≤ 120, 2x + 2y ≤ 100, x ≥ 0, y ≥ 0

• x + y ≤ 100, 2x + y ≤ 120, x ≥ 0, y ≥ 0

• x + y ≤ 80, 2x + y ≤ 100, x ≥ 0, y ≥ 0

A school is organizing a field trip and has a budget of $500. The cost per student for the trip is $20, and the cost for a bus is $200. Write a system of inequalities to represent the number of students that can attend the trip and analyze the feasible solutions.

• 0 <= x <= 15

• 0 <= x <= 10

• 0 <= x <= 5

• 0 <= x <= 20

A company produces two types of gadgets: Type A and Type B. Each Type A gadget requires 3 hours of assembly, and each Type B gadget requires 2 hours. If the company has 30 hours of assembly time available, write a system of inequalities and identify the feasible region for the number of gadgets produced.

• 4x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 3x + y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

A restaurant offers two types of meals: vegetarian and non-vegetarian. The vegetarian meal costs $10, and the non-vegetarian meal costs $15. If the restaurant wants to make at least $300 in a day and can serve no more than 40 meals, write a system of inequalities and analyze the feasible solutions.

• The system of inequalities is: 10x + 15y >= 300, x + y <= 40, x >= 0, y >= 0.

• 10x + 15y <= 300, x + y >= 40, x >= 0, y >= 0

• 10x + 15y >= 200, x + y <= 40, x >= 0, y >= 0

• 10x + 15y >= 300, x + y <= 30, x >= 0, y >= 0

A charity event is selling tickets for adults and children. Adult tickets are $15, and children's tickets are $10. If the goal is to raise at least $600 and no more than 50 tickets can be sold, write a system of inequalities and identify the feasible region.

• 15x + 10y >= 600, x + y <= 50, x >= 0, y >= 0

• 15x + 10y <= 600

• x >= 10, y >= 5

• x + y >= 50

A factory produces two products: X and Y. Each product X requires 4 hours of machine time, and each product Y requires 2 hours. If the factory has 40 hours of machine time available, write a system of inequalities and identify the feasible region for the production of products.

• 3x + 2y ≤ 40, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 0

• 4x + 3y ≤ 40, x ≥ 0, y ≥ 0

A local bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours to bake, and each vanilla cake requires 1 hour. If the bakery has 10 hours available for baking, write a system of inequalities and analyze the feasible solutions.

• The system of inequalities is: 2x + y ≤ 10, x ≥ 0, y ≥ 0.

• 2x + y < 10, x ≥ 0, y ≤ 0

• x + 2y ≤ 10, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 10, x ≥ 0, y ≥ 0

A clothing store sells shirts and pants. Each shirt costs $20, and each pair of pants costs $30. If the store wants to make at least $1000 and can sell no more than 50 items, write a system of inequalities and identify the feasible region.

• The system of inequalities is: 20x + 30y >= 1000, x + y <= 50, x >= 0, y >= 0.

• 20x + 30y <= 1000, x + y >= 50, x >= 0, y >= 0.

• 20x + 30y >= 1000, x + y <= 100, x >= 0, y >= 0.

• 20x + 30y >= 500, x + y <= 50, x >= 0, y >= 0.

A tech company is developing two software applications. Each application requires 5 hours of coding and 3 hours of testing. If the company has 40 hours of coding and 30 hours of testing available, write a system of inequalities and analyze the feasible solutions.

• x + y <= 12; x + y <= 15; x >= 0; y >= 0

• x + y <= 5; x + y <= 8; x >= 0; y >= 0

• x + y <= 8; x + y <= 10; x >= 0; y >= 0

• x + y <= 10; x + y <= 20; x >= 1; y >= 1