Real-Life Inequalities: Identifying Feasible Regions

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 the inequalities that represent the feasible region for the number of acres of corn (x) and wheat (y) he can plant.

• x + y >= 100

• 2x + y >= 120

• x + y <= 100, 2x + y <= 120, x >= 0, y >= 0

• x + 2y <= 100

A school is organizing 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 (x) that can attend the trip, and identify the feasible region.

• 35x ≤ 500, x ≤ 14

• 30x ≤ 500, x ≤ 10

• 40x ≤ 500, x ≤ 12

• 25x ≤ 500, x ≤ 20

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 of flour. If the bakery has 30 cups of flour, write the inequality that represents the number of chocolate (x) and vanilla (y) cakes that can be made.

• 3x + 2y ≤ 30

• 3x + y ≤ 30

• 4x + 2y ≤ 30

• 2x + 3y ≤ 30

A gym has a maximum capacity of 200 members. If 3 times the number of adult members (x) plus 2 times the number of child members (y) must be less than or equal to 200, write the inequality and identify the feasible region for the number of adult and child members.

• x + 5y <= 200

• 2x + 3y <= 200

• 3x + 2y <= 200

• 4x + y <= 200

A concert hall can seat a maximum of 300 people. If tickets for adults cost $15 and tickets for children cost $10, and the total revenue must be at least $2000, write the inequality that represents the relationship between the number of adult (x) and child (y) tickets sold.

• x + y ≥ 300 and 15x + 10y ≤ 2000

• x + y ≤ 250 and 15x + 10y ≥ 1500

• x + y ≤ 300 and 15x + 10y ≥ 2000

• x + y = 300 and 15x + 10y = 2000

A company produces two products, A and B. Each product A requires 4 hours of labor and each product B requires 2 hours of labor. If the company has 40 hours of labor available, write the inequalities that represent the feasible region for the number of products A (x) and B (y) that can be produced.

• 4x + 2y ≤ 40, x ≥ 0, y ≥ 0

• x + y ≤ 20

• 3x + 4y ≤ 40

• 2x + 3y ≤ 40

A restaurant has a maximum of 50 tables. If each table can seat 4 people, write an inequality to represent the maximum number of people (x) that can be seated in the restaurant, and identify the feasible region.

• x <= 100

• x <= 200

• x <= 250

• x <= 150

A charity event aims to raise at least $1000. If each ticket sold for adults (x) raises $25 and each ticket sold for children (y) raises $15, write the inequality that represents the total amount raised and identify the feasible region.

• 20x + 10y ≥ 1000

• 30x + 20y ≤ 1000

• 25x + 25y ≥ 1000

• 25x + 15y ≥ 1000

A factory produces two types of toys: dolls and cars. Each doll requires 2 hours of production time and each car requires 3 hours. If the factory has 24 hours available, write the inequality that represents the maximum number of dolls (x) and cars (y) that can be produced.

• 2x + 3y ≤ 24

• x + 2y ≤ 12

• 4x + 2y ≤ 24

• 2x + 3y < 24

A school is planning a sports day and has a budget of $800. The cost for renting equipment is $50 per team (x) and $30 per participant (y). Write an inequality to represent the maximum number of teams and participants that can be accommodated within the budget, and identify the feasible region.

• 50x + 30y < 800

• 40x + 20y ≤ 800

• 50x + 30y ≤ 800

• 50x + 30y = 800