Real-World Systems of Inequalities: Finding Feasible Regions

A farmer has 100 feet of fencing to create a rectangular pen for sheep. The length of the pen must be at least 10 feet. Set up a system of inequalities to represent the possible dimensions of the pen. What are the feasible regions for the length and width?

• L >= 5 and W <= 60 - L

• L >= 10 and W <= 50 - L

• L <= 10 and W >= 50 - L

• L + W <= 100 and L >= 15

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

• 0 <= x <= 15

• 0 <= x <= 5

• 0 <= x <= 10

• 0 <= x <= 20

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 hours of labor, and each vanilla cake requires 1 hour. If the bakery has 10 hours of labor available, set up a system of inequalities to represent the number of each type of cake that can be made. What is the feasible region for the number of cakes?

• The feasible region is defined by the inequalities: 2x + y <= 10, x >= 0, y >= 0.

• 3x + y <= 10, x >= 0, y >= 0

• 2x + y >= 10, x <= 0, y <= 0

• x + 2y <= 10, x >= 0, y <= 0

A gym offers two types of memberships: basic and premium. The basic membership costs $30 per month, and the premium costs $50. If a family wants to spend no more than $200 per month on memberships, set up a system of inequalities. What combinations of memberships are feasible?

• (2, 3)

• The feasible combinations of memberships are those that satisfy the inequalities, such as (0, 4), (1, 3), (2, 2), (3, 1), (6, 0) for (x, y) respectively.

• (4, 2)

• (0, 5)

A local theater has 300 seats and sells tickets for $10 for adults and $5 for children. If the theater wants to make at least $2000 from ticket sales, set up a system of inequalities to represent the number of adult and child tickets sold. What is the feasible region for ticket sales?

• x + y >= 300, 10x + 5y <= 2000, x <= 0, y <= 0

• x + y <= 250, 10x + 5y >= 1500, x >= 10, y >= 10

• x + y <= 300, 10x + 5y <= 2500, x >= 0, y <= 0

• x + y <= 300, 10x + 5y >= 2000, x >= 0, y >= 0

A clothing store has a sale on shirts and pants. Shirts are $15 each, and pants are $25 each. If a customer wants to spend no more than $100 and buy at least 2 shirts, set up a system of inequalities. What combinations of shirts and pants can the customer buy?

• The customer can buy combinations of shirts and pants such as (2 shirts, 2 pants), (3 shirts, 1 pant), (4 shirts, 0 pants), etc., as long as the total cost does not exceed $100.

• (1 shirt, 3 pants)

• (2 shirts, 4 pants)

• (5 shirts, 0 pants)

A charity event is selling tickets for $25 each and has a goal to raise at least $1500. If they also have a fixed cost of $300 for the venue, set up a system of inequalities to represent the number of tickets they need to sell. What is the feasible region for ticket sales?

• x <= 60 and x >= 0

• x >= 80 and x <= 100

• x >= 72 and x >= 0

• x <= 72 and x >= 0

A car rental company has a fleet of cars and wants to rent them out. Each car can be rented for $40 per day. If the company wants to earn at least $800 in a week, set up a system of inequalities to represent the number of cars rented. What is the feasible region for rentals?

• x >= 3

• x < 3

• x <= 2

• x = 5

A pizza shop sells two sizes of pizzas: small and large. A small pizza costs $8, and a large pizza costs $12. If the shop wants to make at least $300 in a day and sell at least 10 pizzas, set up a system of inequalities. What combinations of small and large pizzas can be sold?

• 8x + 12y <= 300, x + y <= 10, x >= 0, y >= 0.

• 8x + 12y >= 300, x + y <= 10, x >= 0, y >= 0.

• 8x + 12y >= 200, x + y >= 5, x >= 0, y >= 0.

• The system of inequalities is: 8x + 12y >= 300, x + y >= 10, 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. If the store wants to have at least 10 dogs, set up a system of inequalities. What is the feasible region for the number of dogs and cats?

• 2x + y >= 50, x <= 10, x >= 0, y >= 0

• x + y <= 50, x >= 10, y <= 0

• 2x + y <= 40, x >= 5, y >= 0

• The system of inequalities is: 2x + y <= 50, x >= 10, x >= 0, y >= 0.