Identifying Feasible Regions in Real-World Scenarios

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, what are the inequalities that represent the feasible region for planting corn (x) and wheat (y)?

• x + y ≤ 50, 2x + y ≤ 80, x ≥ 0, y ≥ 0

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

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

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

A school is organizing a field trip and has a budget of $500. Each student ticket costs $15, and each adult ticket costs $25. Write the system of inequalities that represents the number of student (x) and adult (y) tickets they can sell without exceeding the budget.

• 10x + 20y ≤ 500, x ≥ 0, y ≥ 0

• 15x + 25y ≤ 500, x ≥ 0, y ≥ 0

• 15x + 30y ≤ 500, x ≥ 0, y ≥ 0

• 20x + 25y ≤ 500, x ≥ 0, y ≥ 0

A company produces two types of gadgets, A and B. Each gadget A requires 3 hours of labor and each gadget B requires 2 hours. The company has 60 hours of labor available. If they want to produce at least 5 gadgets A, what inequalities represent the feasible production region?

• 3x + 2y ≤ 60, y ≥ 5

• x + y ≤ 60, x ≥ 5

• 3x + 2y ≤ 60, x ≥ 5

• 3x + 2y ≥ 60, x ≥ 5

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 pounds of flour and each vanilla cake requires 1 pound. If the bakery has 20 pounds of flour and wants to make at least 5 chocolate cakes, what inequalities represent the feasible region for the number of chocolate (x) and vanilla (y) cakes?

• x + 2y ≤ 20, x ≥ 3

• 2x + 3y ≤ 20, x ≥ 5

• x + y ≤ 20, y ≥ 5

• 2x + y ≤ 20, x ≥ 5

A gym has a maximum capacity of 200 members. If they currently have 150 members and want to add new members, how many new members (x) can they add if they want to keep the total under 200? Write the inequality that represents this situation.

• x < 30

• x < 200

• x < 100

• x < 50

A concert hall has 300 seats. If tickets for adults cost $20 and tickets for children cost $10, and the total revenue must be at least $4000, what inequalities can represent the number of adult (x) and child (y) tickets sold?

• x + y >= 300, 20x + 10y <= 4000

• x + y = 300, 20x + 10y = 4000

• x + y <= 250, 20x + 10y >= 5000

• x + y <= 300, 20x + 10y >= 4000

A local charity is organizing a fundraiser. They plan to sell two types of items: T-shirts and mugs. Each T-shirt costs $10 and each mug costs $5. If they want to raise at least $300, what inequalities represent the number of T-shirts (x) and mugs (y) they need to sell?

• 10x + 5y = 300, x ≥ 0, y ≥ 0

• 5x + 10y ≥ 300, x ≥ 0, y ≥ 0

• 10x + 5y ≥ 300, x ≥ 0, y ≥ 0

• 10x + 5y ≤ 300, x ≥ 0, y ≥ 0

A restaurant has a seating capacity of 150. If they have 80 tables and each table can seat 4 people, how many additional tables (x) can they add without exceeding the seating capacity? Write the inequality that represents this situation.

• 25

• 60

• 50

• 37

A factory produces two products, P1 and P2. Each product P1 requires 4 hours of machine time and each product P2 requires 3 hours. If the factory has 120 hours of machine time available, what inequalities represent the feasible production region for products P1 (x) and P2 (y)?

• 5x + 3y ≤ 120, x ≥ 0, y ≥ 0

• 4x + 3y ≤ 120, x ≥ 0, y ≥ 0

• 2x + 4y ≤ 120, x ≥ 0, y ≥ 0

• 4x + 2y ≤ 120, x ≥ 0, y ≥ 0

A clothing store sells shirts and pants. Each shirt costs $15 and each pair of pants costs $25. If the store wants to make at least $1000 in sales, what inequalities represent the number of shirts (x) and pants (y) they need to sell?

• 15x + 25y ≤ 1000

• 10x + 20y ≥ 1000

• 15x + 25y ≥ 1000

• 20x + 30y ≥ 1000