Graphing Inequalities: Finding Feasible Regions in Context

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, graph the inequalities and identify the feasible region for planting corn (C) and wheat (W).

• The feasible region is defined by the inequalities C + W <= 100 and 2C + W <= 120, with C, W >= 0.

• C + W = 100 and 2C + W = 120

• C + W <= 80 and 2C + W <= 100

• C + W >= 100 and 2C + W >= 120

A school is organizing a field trip and has a budget of $500. The cost per student for the trip is $20, and the bus rental costs $200. Write the inequalities representing the number of students (S) that can attend and graph them to find the feasible region.

• S <= 10

• S <= 20

• S <= 5

• S <= 15

A company produces two products, A and B. Each product A requires 3 hours of machine time, and each product B requires 2 hours. The company has a maximum of 30 hours of machine time available. Graph the inequalities for the production of A and B and identify the feasible region.

• The feasible region is the area bounded by the axes and the line 3x + 2y = 15.

• The feasible region is the area below the axes in the first quadrant.

• The feasible region is the area above the line 3x + 2y = 30.

• The feasible region is the area bounded by the axes and the line 3x + 2y = 30 in the first quadrant.

A restaurant can serve a maximum of 200 meals in a day. Each meal requires 1 hour of cooking time, and the restaurant has 150 hours of cooking time available. Write the inequalities for the number of meals served (M) and graph them to find the feasible region.

• M <= 150 and M <= 200

• M >= 150 and M >= 200

• M = 150 or M = 200

• M <= 100 and M <= 250

A charity event is selling tickets for adults and children. Adult tickets cost $15, and children's tickets cost $10. The total revenue must be at least $300, and they can sell a maximum of 50 tickets. Graph the inequalities and identify the feasible region for adult (A) and child (C) tickets sold.

• The feasible region is defined by the inequalities 10A + 15C >= 300 and A + C <= 50, with A, C >= 0.

• The feasible region is defined by the inequalities 20A + 5C >= 300 and A + C <= 50, with A, C >= 0.

• The feasible region is defined by the inequalities 15A + 10C <= 300 and A + C >= 50, with A, C >= 0.

• The feasible region is defined by the inequalities 15A + 10C >= 300 and A + C <= 50, with A, C >= 0.

A gym has a maximum capacity of 150 members. Each membership costs $30, and the gym wants to earn at least $3000 in membership fees. Write the inequalities for the number of adult (X) and student (Y) memberships and graph them to find the feasible region.

• X + Y ≤ 150 and X + Y ≥ 100

• X + Y ≤ 100 and X + Y ≥ 150

• X + Y ≤ 200 and X + Y ≥ 50

• X + Y ≤ 150 and X + Y ≥ 300

A factory produces two types of toys: dolls and cars. Each doll requires 4 hours of labor, and each car requires 2 hours. The factory has a total of 40 hours of labor available. Graph the inequalities for the production of dolls (D) and cars (C) and identify the feasible region.

• D + 2C ≤ 30, D ≥ 0, C ≥ 0

• D + C ≤ 25, D ≥ 0, C ≥ 0

• 2D + C ≤ 10, D ≥ 0, C ≥ 0

• The feasible region is defined by the inequalities: 2D + C ≤ 20, D ≥ 0, C ≥ 0.

A local bakery can produce a maximum of 200 loaves of bread and pastries combined. Each loaf of bread takes 1 hour to bake, and each pastry takes 0.5 hours. If the bakery has 100 hours available, graph the inequalities and identify the feasible region for loaves (L) and pastries (P).

• L + P ≤ 250 and L + P ≥ 100

• The feasible region is defined by the inequalities L + P ≤ 200 and L + 0.5P ≤ 100, with L, P ≥ 0.

• L + P ≤ 150 and L + 0.5P ≤ 80

• L + P ≤ 200 and L + 0.5P ≤ 120

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 $600 in sales and can sell a maximum of 40 items. Write the inequalities for shirts (S) and pants (P) and graph them to find the feasible region.

• The inequalities are: 20S + 30P >= 600, S + P <= 40, S >= 0, P >= 0.

• 20S + 30P <= 600, S + P >= 40, S >= 0, P >= 0

• 10S + 20P >= 600, S + P <= 50, S >= 0, P >= 0

• 20S + 30P >= 600, S + P >= 40, S >= 0, P <= 0

A tech company is developing two products, X and Y. Each product X requires 5 hours of development time, and each product Y requires 3 hours. The company has a total of 60 hours available for development. Graph the inequalities for the production of X and Y and identify the feasible region.

• The feasible region includes negative values for x and y.

• The feasible region is unbounded in the first quadrant.

• The line 5x + 3y = 60 intersects the y-axis at 20.

• The feasible region is bounded by the axes and the line 5x + 3y = 60 in the first quadrant.