Graphing Inequalities: Finding Feasible Regions in Real Life

A farmer has a rectangular field. The length of the field is 3 meters more than twice the width. If the perimeter of the field must be less than 60 meters, graph the inequalities and identify the feasible region for the dimensions of the field.

• The feasible region for the dimensions of the field is 0 < w < 12 and 3 < l < 25.

• The feasible region for the dimensions of the field is 0 < w < 5 and 3 < l < 15.

• The feasible region for the dimensions of the field is 0 < w < 10 and 3 < l < 18.

• The feasible region for the dimensions of the field is 0 < w < 9 and 3 < l < 21.

A school is planning to organize a sports event. They can spend no more than $500 on equipment and $300 on refreshments. If the cost of equipment is $50 per set and refreshments cost $15 per person, graph the system of inequalities and find the feasible region for the number of sets and people.

• The feasible region is defined by the points (0,0), (12,0), (0,10), and (12,10).

• The feasible region is defined by the points (0,0), (10,0), (0,20), and (10,20).

• The feasible region is defined by the points (0,0), (8,0), (0,25), and (8,25).

• The feasible region is defined by the points (0,0), (5,0), (0,15), and (5,15).

A company produces two types of gadgets. Each gadget A requires 2 hours of labor and $30 in materials, while gadget B requires 3 hours of labor and $20 in materials. If the company has a maximum of 60 hours of labor and $300 available, graph the inequalities and identify the feasible region for the production of gadgets A and B.

• The feasible region is defined by the inequalities 2x + 2y ≤ 60 and 30x + 10y ≤ 300.

• The feasible region is defined by the inequalities 2x + 3y ≤ 60 and 30x + 20y ≤ 300, along with x ≥ 0 and y ≥ 0.

• The feasible region is defined by the inequalities 3x + 2y ≤ 60 and 20x + 30y ≤ 300.

• The feasible region is defined by the inequalities 2x + 3y ≤ 30 and 30x + 20y ≤ 150.

A restaurant offers two types of meals: vegetarian and non-vegetarian. The restaurant can serve a maximum of 100 meals, and they want to ensure that at least 30 meals are vegetarian. If each vegetarian meal costs $10 and each non-vegetarian meal costs $15, graph the inequalities and find the feasible region for the number of each type of meal they can serve.

• The feasible region is defined by the vertices (30, 30), (100, 30), (30, 100), and (70, 70).

• The feasible region is defined by the vertices (0, 30), (0, 100), (30, 100), and (70, 0).

• The feasible region is defined by the vertices (10, 10), (90, 10), (50, 50), and (10, 50).

• The feasible region is defined by the vertices (30, 0), (100, 0), (70, 30), and (30, 70).

A local gym has a maximum capacity of 200 members. They want to maintain at least 50 members in their yoga classes. If each yoga class can accommodate 10 members and each fitness class can accommodate 20 members, graph the inequalities and identify the feasible region for the number of yoga and fitness classes they can offer.

• The feasible region is defined by the inequalities 10x + 20y <= 200 and x >= 5, with x, y >= 0.

• The feasible region is defined by the inequalities 10x + 20y <= 200 and x >= 15, with x, y >= 0.

• The feasible region is defined by the inequalities 10x + 20y <= 150 and x >= 10, with x, y >= 0.

• The feasible region is defined by the inequalities 10x + 20y <= 250 and x >= 0, with x, y >= 0.

A clothing store sells shirts and pants. Each shirt costs $20 and each pair of pants costs $30. The store has a budget of $600 for inventory and wants to stock at least 10 shirts. Graph the inequalities and find the feasible region for the number of shirts and pants they can purchase.

• The feasible region is defined by the inequalities 20x + 30y >= 600 and x <= 10.

• The feasible region is defined by the inequalities 20x + 30y <= 600 and x >= 10, with x and y being non-negative.

• The store can purchase a maximum of 20 shirts and 5 pairs of pants.

• The inequalities are 20x + 30y = 600 and x = 10.

A charity organization is planning a fundraising event. They can spend no more than $800 on venue and catering. The venue costs $200 and catering costs $50 per person. Graph the system of inequalities and identify the feasible region for the number of people they can invite to the event.

• The charity can invite up to 12 people.

• The charity can invite up to 20 people.

• The charity can invite up to 10 people.

• The charity can invite up to 15 people.

A tech startup is developing two products. Product X requires 4 hours of development and $100 in materials, while Product Y requires 2 hours of development and $50 in materials. If they have a maximum of 40 hours of development time and $500 available, graph the inequalities and find the feasible region for the production of products X and Y.

• The feasible region is defined by the inequalities 3x + 2y <= 15 and x + y <= 25, with intersection points (0,0), (0,15), (5,5), and (10,0).

• The feasible region is defined by the inequalities 2x + 3y <= 30 and x + y <= 20, with intersection points (0,0), (0,10), (15,0), and (10,5).

• The feasible region is defined by the inequalities x + 2y <= 20 and 4x + y <= 40, with intersection points (0,0), (0,20), (10,0), and (5,10).

• The feasible region is defined by the inequalities 2x + y <= 10 and 2x + y <= 20, with the intersection points being (0,0), (0,10), (5,0), and (10,0).

A bakery sells two types of cakes: chocolate and vanilla. Each chocolate cake requires 3 hours of baking and $15 in ingredients, while each vanilla cake requires 2 hours of baking and $10 in ingredients. If the bakery has a maximum of 30 hours of baking time and $300 available, graph the inequalities and identify the feasible region for the number of each type of cake they can bake.

• The feasible region is defined by the inequalities 3x + 2y <= 30 and 15x + 10y <= 300, with x >= 0 and y >= 0.

• The feasible region is defined by the inequalities 4x + y <= 30 and 5x + 20y <= 300.

• The feasible region is defined by the inequalities 2x + 3y <= 30 and 10x + 15y <= 300.

• The feasible region is defined by the inequalities 3x + 5y <= 30 and 20x + 10y <= 300.

A concert venue can accommodate a maximum of 500 people. They want to ensure that at least 200 tickets are sold for the VIP section. If VIP tickets cost $100 each and regular tickets cost $50 each, graph the inequalities and find the feasible region for the number of VIP and regular tickets they can sell.

• The feasible region is defined by the vertices of the inequalities: (250, 250), (400, 100), and (0, 0).

• The feasible region is defined by the vertices of the inequalities: (100, 400), (300, 200), and (0, 500).

• The feasible region is defined by the vertices of the inequalities: (200, 300), (500, 0), and (200, 0).

• The feasible region is defined by the vertices of the inequalities: (150, 350), (450, 50), and (0, 100).