Graphing and Analyzing Intersections of Nonlinear Inequalities

A farmer wants to plant two types of crops, A and B. Crop A requires at least 3 hours of sunlight and 2 units of water per week, while crop B requires at least 2 hours of sunlight and 3 units of water. If the farmer has a maximum of 12 hours of sunlight and 15 units of water available, graph the system of inequalities and determine the feasible region for planting both crops.

• The feasible region is bounded by the lines 3x + 2y = 12 and 2x + 3y = 15, representing the maximum sunlight and water constraints.

• The feasible region is defined by the lines 2x + 2y = 12 and 3x + 3y = 15.

• The feasible region is represented by the lines 3x + 3y = 12 and 2x + 2y = 15.

• The feasible region is unbounded due to unlimited sunlight and water.

A company produces two products, X and Y. Product X requires 4 hours of labor and 2 units of raw material, while product Y requires 3 hours of labor and 5 units of raw material. If the company has a maximum of 24 hours of labor and 30 units of raw material available, graph the system of inequalities and analyze the intersection points to find the optimal production plan.

• Produce 2 units of product X and 5 units of product Y.

• Produce 3 units of product X and 4 units of product Y.

• Produce 1 unit of product X and 6 units of product Y.

• Produce 4 units of product X and 2 units of product Y.

A park is designed with two types of recreational areas: playgrounds and picnic areas. Each playground requires 5,000 square feet and each picnic area requires 3,000 square feet. The total area available for development is 30,000 square feet. Graph the inequalities representing the area constraints and identify the feasible region for the park's layout.

• The feasible region is defined by the inequalities: 5000x + 3000y ≤ 30000, x ≥ 0, y ≥ 0.

• 5000x + 3000y ≥ 30000, x ≤ 0, y ≤ 0

• 5000x + 3000y = 30000, x ≥ 0, y ≤ 0

• 5000x + 3000y ≤ 25000, x ≥ 0, y ≥ 0

A school is organizing a field trip and has a budget for transportation and food. The cost for transportation is $200 per bus and $10 per student for food. If the school has a budget of $1,000, graph the system of inequalities and determine the maximum number of students that can attend the trip while staying within budget.

• 100

• 120

• 150

• 80

A bakery produces two types of cakes: chocolate and vanilla. Each chocolate cake requires 2 cups of flour and 1 cup of sugar, while each vanilla cake requires 1 cup of flour and 2 cups of sugar. If the bakery has 10 cups of flour and 8 cups of sugar, graph the inequalities and find the feasible region for the number of cakes that can be made.

• The feasible region is defined by the vertices (0,0), (0,4), (5,0), and (2,3).

• The feasible region is defined by the vertices (1,1), (0,5), (4,2), and (3,2).

• The feasible region is defined by the vertices (2,2), (0,3), (3,1), and (1,4).

• The feasible region is defined by the vertices (0,0), (0,8), (10,0), and (5,1).

A local gym offers two types of fitness classes: yoga and spinning. Each yoga class requires 1 instructor and can accommodate 10 participants, while each spinning class requires 2 instructors and can accommodate 15 participants. If the gym has 5 instructors and a maximum of 60 participants, graph the system of inequalities and analyze the feasible combinations of classes.

• (1 yoga, 1 spinning)

• (0 yoga, 4 spinning)

• The feasible combinations of classes are (0 yoga, 3 spinning), (2 yoga, 0 spinning), and (1 yoga, 2 spinning).

• (3 yoga, 0 spinning)

A tech company is developing two software products, A and B. Product A requires 3 hours of coding and 2 hours of testing, while product B requires 2 hours of coding and 3 hours of testing. If the company has a total of 18 hours for coding and 12 hours for testing, graph the inequalities and determine the feasible region for product development.

• The feasible region is defined by the inequalities 3x + 3y <= 18 and 2x + 2y <= 12.

• The feasible region is defined by the inequalities 3x + 2y <= 18 and 2x + 3y <= 12, along with x >= 0 and y >= 0.

• The feasible region is defined by the inequalities 3x + 2y <= 12 and 2x + 3y <= 18.

• The feasible region is defined by the inequalities 3x + 2y >= 18 and 2x + 3y >= 12.

A restaurant offers two types of meal packages: vegetarian and non-vegetarian. Each vegetarian package requires 2 pounds of vegetables and 1 pound of grains, while each non-vegetarian package requires 3 pounds of meat and 2 pounds of grains. If the restaurant has 20 pounds of vegetables, 15 pounds of meat, and 25 pounds of grains, graph the system of inequalities and find the feasible meal combinations.

• (2, 3)

• Feasible meal combinations: (0, 0), (0, 5), (10, 0), (5, 10)

• (6, 6)

• (4, 8)

A clothing store sells two types of shirts: casual and formal. Each casual shirt requires 2 yards of fabric and 1 hour of labor, while each formal shirt requires 3 yards of fabric and 2 hours of labor. If the store has 30 yards of fabric and 20 hours of labor available, graph the inequalities and analyze the feasible production options.

• The feasible production options are determined by the vertices of the feasible region, which can be calculated as (0,0), (0,10), (15,0), and (6,7).

• The feasible production options are (0,0), (20,0), (0,10), and (10,10).

• The feasible production options are (5,5), (10,5), (0,20), and (15,5).

• The feasible production options are (0,0), (10,0), (0,15), and (7,6).

A community garden has space for two types of plants: flowers and vegetables. Each flower bed requires 4 square feet and each vegetable bed requires 6 square feet. If the total area available for planting is 48 square feet, graph the system of inequalities and determine the maximum number of flower and vegetable beds that can be planted.

• Maximum number of flower and vegetable beds is 10 (6 flowers and 4 vegetables).

• Maximum number of flower and vegetable beds is 12 (8 flowers and 4 vegetables).

• Maximum number of flower and vegetable beds is 8 (4 flowers and 4 vegetables).

• Maximum number of flower and vegetable beds is 9 (5 flowers and 4 vegetables).