Exploring Feasible Regions in Non-Linear Inequalities

A farmer has a rectangular field where the length is twice the width. If the area of the field must be less than 2000 square meters, graph the system of inequalities that represents this situation and identify the feasible region.

• The feasible region is defined by 0 < w < √1000 and length = 2w.

• The feasible region is defined by 0 < w < 500 and length = 3w.

• The feasible region is defined by w > 0 and length = w.

• The area of the field must be greater than 2000 square meters.

A company produces two types of products, A and B. The profit from product A is $5 per unit, and from product B is $8 per unit. The production constraints are given by the inequalities x^2 + y^2 ≤ 100 and x + 2y ≤ 40. Graph the system and determine the maximum profit region.

• Maximum profit is $50 at (10, 0).

• Maximum profit is $30 at (5, 10).

• Maximum profit is $40 at (8, 8).

• Maximum profit is $64 at (0, 20).

A park is designed in the shape of a circular area with a radius of 10 meters. There is also a rectangular area adjacent to it where the length is 3 meters longer than the width. If the total area of both sections must be less than 400 square meters, graph the inequalities and find the intersection of the feasible regions.

• The feasible region for the width w is approximately 0 < w < 10.5 meters.

• The feasible region for the width w is approximately 10 < w < 15 meters.

• The feasible region for the width w is approximately 0 < w < 20 meters.

• The feasible region for the width w is approximately 0 < w < 5 meters.

A school is planning to organize a sports event. The number of participants in soccer (x) and basketball (y) must satisfy the inequalities x^2 + y ≤ 50 and 2x + y^2 ≤ 100. Graph the system and analyze the intersection points to determine the possible number of participants.

• Maximum participants: 8

• Maximum participants: 15

• Maximum participants: 5

• Maximum participants: 12

A local bakery sells two types of cakes: chocolate and vanilla. The number of chocolate cakes (x) and vanilla cakes (y) must satisfy the inequalities x + 2y ≤ 30 and x^2 + y^2 ≤ 100. Graph the inequalities and find the feasible region for cake production.

• The feasible region is only defined by x + 2y ≤ 30.

• The feasible region is the area where x^2 + y^2 ≤ 100 is satisfied alone.

• The feasible region is the area where x + 2y ≤ 30 and x^2 + y^2 ≤ 100 overlap.

• The feasible region includes all points where x + y ≤ 30.

A car rental company has a fleet of cars that can be rented for a maximum of 50 hours per week. The rental hours for economy cars (x) and luxury cars (y) must satisfy the inequalities x + y ≤ 50 and x^2 + 2y ≤ 100. Graph the system and identify the feasible rental hours.

• The feasible rental hours are only for economy cars.

• The maximum rental hours for luxury cars is 50 hours.

• The feasible rental hours are the points (x, y) that lie outside the overlapping region of the two inequalities.

• The feasible rental hours are the points (x, y) that lie within the overlapping region of the two inequalities.

A restaurant offers two types of meals: vegetarian (x) and non-vegetarian (y). The total number of meals must be less than 80, and the cost constraints are given by the inequalities 3x + 5y ≤ 300 and x^2 + y^2 ≤ 400. Graph the inequalities and analyze the intersection for meal planning.

• 50 vegetarian meals and 10 non-vegetarian meals.

• 30 vegetarian meals and 20 non-vegetarian meals.

• 10 vegetarian meals and 30 non-vegetarian meals.

• The optimal meal combination is to serve 40 vegetarian meals and 0 non-vegetarian meals.

A tech company is developing two software products. The development time for product A (x) and product B (y) must satisfy the inequalities x + 3y ≤ 60 and 2x^2 + y ≤ 100. Graph the system and determine the feasible development time for both products.

• The development time for both products is unlimited.

• The feasible development time for products A and B is the region defined by the inequalities, specifically the area where both conditions are satisfied, typically found by evaluating the vertices of the intersection of the lines.

• The feasible development time is only for product A.

• The inequalities do not define a feasible region.

A clothing store sells shirts and pants. The number of shirts (x) and pants (y) must satisfy the inequalities 2x + y ≤ 50 and x^2 + 3y ≤ 150. Graph the inequalities and find the intersection points to determine the maximum inventory.

• Maximum inventory is 30 items (10 shirts and 20 pants).

• Maximum inventory is 35 items (15 shirts and 20 pants).

• Maximum inventory is 25 items (15 shirts and 10 pants).

• Maximum inventory is 40 items (20 shirts and 20 pants).

A fitness center offers two types of memberships: basic (x) and premium (y). The total number of memberships sold must be less than 100, and the revenue constraints are given by the inequalities 4x + 6y ≤ 400 and x^2 + y^2 ≤ 200. Graph the system and analyze the feasible membership combinations.

• The feasible combinations of memberships (x, y) are those that satisfy all three inequalities, such as (0, 66), (50, 0), (20, 30), etc.

• (10, 70)

• (30, 50)

• (100, 0)