Real-World Applications: Identifying Feasible Regions in Inequalities

A farmer has 100 meters of fencing to create a rectangular pen for sheep. If the length of the pen is represented by x and the width by y, write a system of inequalities to represent the constraints on the dimensions of the pen. Identify the feasible region.

• x + y = 100, 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 a system of inequalities to represent the maximum number of students that can attend. What is the feasible region for the number of students?

• x ≥ 0 and x ≤ 20; feasible region: {0, 1, 2, ..., 20}

• x ≥ 0 and x ≤ 10; feasible region: {0, 1, 2, ..., 10}

• x ≥ 0 and x ≤ 5; feasible region: {0, 1, 2, ..., 5}

• x ≥ 0 and x ≤ 14; feasible region: {0, 1, 2, ..., 14}

A factory produces two types of toys: A and B. Each toy A requires 2 hours of labor and each toy B requires 3 hours. The factory has a maximum of 30 hours of labor available per week. Write a system of inequalities to represent the production limits. Identify the feasible region for the number of toys produced.

• x + y ≤ 30, x ≥ 0, y ≥ 0

• 2x + y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 2x + 3y ≥ 30, x ≥ 0, y ≥ 0

A gym offers two types of memberships: basic and premium. The basic membership costs $25 per month, and the premium membership costs $40 per month. If the gym wants to earn at least $1,000 in a month, write a system of inequalities to represent the number of each type of membership sold. What is the feasible region?

• 25x + 40y >= 1000, x >= 0, y >= 0

• 30x + 35y >= 1000, x >= 0, y >= 0

• 25x + 40y <= 1000, x >= 0, y >= 0

• 25x + 40y = 1000, 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 30 cups of flour available, write a system of inequalities to represent the maximum number of cakes that can be made. Identify the feasible region for the cakes.

• 2x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 3y ≤ 30, x ≥ 0, y ≥ 0

• 3x + 2y ≤ 30, x ≥ 0, y ≥ 0

• 4x + y ≤ 30, x ≥ 0, y ≥ 0

A concert hall has a seating capacity of 500. Tickets for the front row cost $50 each, and tickets for the back row cost $30 each. If the concert hall wants to make at least $15,000 from ticket sales, write a system of inequalities to represent the ticket sales. What is the feasible region for the number of tickets sold?

• x + y = 500, 50x + 30y = 15000

• x + y ≥ 500, 50x + 30y ≤ 15000

• x + y ≤ 500, 50x + 30y ≥ 15000

• x + y ≤ 400, 50x + 30y ≥ 20000

A local charity is organizing a fundraiser and has a goal of raising at least $2,000. They plan to sell two types of items: T-shirts for $15 each and mugs for $10 each. Write a system of inequalities to represent the fundraising goal. Identify the feasible region for the number of T-shirts and mugs sold.

• 15x + 10y <= 2000, x >= 0, y >= 0

• 15x + 10y >= 2000, x >= 0, y >= 0

• 15x + 10y = 2000, x >= 0, y >= 0

• 15x + 10y >= 1000, x >= 0, y >= 0

A pet store has a limit of 50 animals it can house. They have dogs and cats, with each dog taking up 2 spaces and each cat taking up 1 space. Write a system of inequalities to represent the maximum number of dogs and cats the store can have. What is the feasible region for the animals?

• 2x + y <= 50, x >= 0, y >= 0

• x + y <= 50

• 3x + y <= 50

• x + 2y <= 50

A clothing store sells shirts and pants. Each shirt costs $20 and each pair of pants costs $30. The store wants to make at least $1,200 in sales. Write a system of inequalities to represent the sales goal. Identify the feasible region for the number of shirts and pants sold.

• 15x + 25y >= 1200, x >= 0, y >= 0

• 20x + 30y >= 1200, x >= 0, y >= 0

• 10x + 20y >= 1200, x >= 0, y >= 0

• 20x + 30y <= 1200, x >= 0, y >= 0

A community center is planning a workshop and has a budget of $800. The cost for materials is $10 per participant and the venue costs $200. Write a system of inequalities to represent the maximum number of participants that can attend. What is the feasible region for the workshop?

• x ≥ 10 and x ≤ 50

• x ≥ 0 and x ≤ 40

• x ≥ 0 and x ≤ 60

• x ≥ 0 and x ≤ 80