Algebraic Solutions and Graphical Interpretations of Inequalities

A farmer has a rectangular field and wants to plant two types of crops. The area for crop A is represented by the inequality x^2 + y^2 ≤ 100, while the area for crop B is represented by the inequality y ≥ x + 10. Graph the system of inequalities and determine the feasible region for planting both crops.

• The feasible region is the area inside the circle x^2 + y^2 ≤ 100 only.

• The feasible region is the area above the line y = x + 10 that lies outside the circle x^2 + y^2 ≤ 100.

• The feasible region is the area above the line y = x + 10 that lies within the circle x^2 + y^2 ≤ 100.

• The feasible region is the area below the line y = x + 10 that lies outside the circle x^2 + y^2 ≤ 100.

A company produces two products, P and Q. The profit from product P is represented by the inequality 3x + 2y ≤ 60, and the production cost for product Q is represented by the inequality y ≤ -x^2 + 20. Solve the inequalities algebraically and interpret the solution graphically to find the maximum profit region.

• The maximum profit region is determined solely by the profit from product P.

• The maximum profit region is defined by the intersection points of the line and the parabola, which can be calculated to find the optimal production levels for products P and Q.

• The inequalities indicate that product Q cannot be produced at all.

• The maximum profit region is a single point where x and y are both zero.

A park is designed with a circular pond represented by the inequality x^2 + y^2 ≤ 36 and a rectangular playground represented by the inequality 0 ≤ x ≤ 6 and 0 ≤ y ≤ 4. Graph the inequalities and find the area available for other park activities.

• 50 square units

• 89.04 square units

• 72 square units

• 100 square units

A school is organizing a fundraiser with two types of events: a bake sale and a car wash. The bake sale can be represented by the inequality 2x + y ≤ 30, while the car wash is represented by y ≤ -0.5x + 15. Graph the system and determine the maximum number of events that can be held.

• 20

• 15

• 30

• 25

A local artist wants to create a mural that fits within a triangular area defined by the inequalities y ≤ -x + 5, y ≥ 0, and x ≥ 0. Solve the inequalities and interpret the graphical solution to find the area available for the mural.

• The area available for the mural is 10 square units.

• The area available for the mural is 8 square units.

• The area available for the mural is 12.5 square units.

• The area available for the mural is 15 square units.

A delivery service has a maximum weight limit for packages. The weight of package A is represented by the inequality 2x + 3y ≤ 30, and the weight of package B is represented by the inequality y ≤ -x^2 + 10. Graph the inequalities and find the feasible weight combinations for the packages.

• The weight of package A is always greater than the weight of package B.

• The feasible weight combinations for packages A and B are the points (x, y) that lie within the overlapping area of the two inequalities.

• The feasible weight combinations are only the points on the boundary of the inequalities.

• There are no feasible weight combinations for the packages.

A restaurant offers two types of meals: vegetarian and non-vegetarian. The vegetarian meals are represented by the inequality x^2 + y^2 ≤ 25, while the non-vegetarian meals are represented by the inequality y ≤ 2x + 5. Graph the system and determine the combinations of meals that can be served.

• The combinations of meals that can be served are the points in the overlapping region of the circle x^2 + y^2 ≤ 25 and the area below the line y ≤ 2x + 5.

• Non-vegetarian meals are represented by the inequality x^2 + y^2 ≤ 25.

• The meals served are only the points outside the circle and above the line.

• The combinations of meals are only vegetarian meals within the circle.

A city is planning to build a new sports complex. The area for basketball courts is represented by the inequality x^2 + (y - 4)^2 ≤ 16, and the area for soccer fields is represented by the inequality y ≤ -0.5x + 10. Solve the inequalities and interpret the solution graphically to find the area for both sports.

• The area for both sports is the region outside the circle x^2 + (y - 4)^2 ≤ 16 that lies below the line y = -0.5x + 10.

• The area for both sports is the region outside the circle x^2 + (y - 4)^2 ≤ 16 that lies above the line y = -0.5x + 10.

• The area for both sports is the region inside the circle x^2 + (y - 4)^2 ≤ 16 that lies below the line y = -0.5x + 10.

• The area for both sports is the region inside the circle x^2 + (y - 4)^2 ≤ 16 that lies above the line y = -0.5x + 10.

A tech company is developing two software products. The development time for product A is represented by the inequality 4x + 3y ≤ 48, and the testing time for product B is represented by the inequality y ≤ -x^2 + 12. Graph the system of inequalities and find the feasible development and testing time.

• The development time for product A is represented by y ≤ 4x + 3.

• The feasible region is bounded by the lines and the parabola, with key points at (0, 12), (12, 0), and intersections of the two curves.

• The feasible region is a straight line with no intersections.

• The key points are (0, 0), (6, 6), and (12, 12).

A community garden has two sections: one for vegetables and one for flowers. The vegetable section is represented by the inequality x^2 + y^2 ≤ 49, while the flower section is represented by the inequality y ≥ 2x - 5. Graph the inequalities and determine the area available for planting both types of plants.

• The area available for planting both types of plants is 49 square units.

• The area available for planting both types of plants is approximately 49π - area below the line.

• The area available for planting both types of plants is 25π.

• The area available for planting both types of plants is 0.