Maximizing Resources: Linear Inequalities & Feasible Regions

A farmer has 100 meters of fencing to create a rectangular pen for his sheep. If the length of the pen is represented by x and the width by y, write the inequality that represents the maximum area of the pen. What is the feasible region for the dimensions of the pen?

• x + y <= 50, x >= 0, y >= 0

• x + y >= 50, x >= 0, y >= 0

• x + y <= 100, x >= 0, y >= 0

• x + y <= 50, x < 0, y >= 0

A school is planning a field trip and has a budget of $500. The cost per student is $20 for transportation and $15 for admission. Write an inequality to represent the maximum number of students that can attend the trip. Identify the feasible region for the number of students.

• x ≤ 20

• x ≤ 10

• x ≤ 14

• x ≤ 12

A factory produces two types of toys: type A and type B. Each type A toy requires 2 hours of labor and each type B toy requires 3 hours. If the factory has 60 hours of labor available, write the inequality that represents the production limits. What is the feasible region for the number of toys produced?

• 2x + 3y ≥ 60

• 3x + 2y ≤ 60

• x + 2y ≤ 60

• 2x + 3y ≤ 60

A gym offers two types of memberships: basic and premium. The basic membership costs $30 per month, and the premium membership costs $50 per month. If a customer wants to spend no more than $200 on memberships, write an inequality to represent this situation. What is the feasible region for the number of each type of membership?

• 30x + 50y = 200, x ≥ 0, y ≥ 0

• 30x + 50y < 200, x ≥ 0, y ≥ 0

• 30x + 50y ≤ 250, x ≥ 0, y ≥ 0

• 30x + 50y ≤ 200, x ≥ 0, y ≥ 0

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 cups of flour and each vanilla cake requires 2 cups. If the bakery has 24 cups of flour, write the inequality that represents the maximum number of cakes that can be made. Identify the feasible region for the number of cakes.

• 3x + 2y ≤ 24

• 2x + 3y ≤ 24

• 4x + y ≤ 24

• x + 4y ≤ 24

A concert hall has a seating capacity of 300. If tickets for the front row cost $50 and tickets for the back row cost $30, and the total revenue must be at least $10,000, write an inequality to represent this situation. What is the feasible region for the number of tickets sold?

• x + y ≤ 250, 50x + 30y ≥ 12000

• x + y ≥ 300, 50x + 30y ≤ 10000

• x + y = 300, 50x + 30y = 10000

• x + y ≤ 300, 50x + 30y ≥ 10000

A company produces two products, A and B. Each product A requires 4 hours of machine time and each product B requires 2 hours. If the company has 40 hours of machine time available, write the inequality that represents the production limits. Identify the feasible region for the number of products produced.

• 3x + 2y ≤ 40

• 4x + 2y ≥ 40

• 4x + 3y ≤ 40

• 4x + 2y ≤ 40

A charity event is selling two types of tickets: VIP tickets for $100 and regular tickets for $50. If the total revenue from ticket sales must be at least $5,000, write an inequality to represent this situation. What is the feasible region for the number of each type of ticket sold?

• 100x + 50y >= 5000

• 50x + 100y >= 5000

• 100x + 50y = 5000

• 100x + 50y <= 5000

A restaurant has a budget of $300 for purchasing ingredients. If the cost of meat is $10 per pound and vegetables are $5 per pound, write an inequality to represent the maximum amount of ingredients that can be purchased. Identify the feasible region for the quantities of meat and vegetables.

• 15x + 3y ≤ 300

• 10x + 5y = 300

• 10x + 5y < 300

• 10x + 5y ≤ 300

A clothing store sells shirts for $25 and pants for $40. If the store wants to make at least $1,000 in sales, write an inequality to represent the sales goal. What is the feasible region for the number of shirts and pants sold?

• 25x + 40y >= 500; x >= 0; y >= 0

• 25x + 40y <= 1000; x >= 0; y >= 0

• 25x + 40y = 1000; x < 0; y >= 0

• 25x + 40y >= 1000; x >= 0; y >= 0