Linear Programming

The feasible region is the triangle with vertices where the lines meet: , , . A linear objective hits its max at a vertex. Evaluate : . . . So the maximum is at .

Since has a negative coefficient in , we want the smallest (take ) and the largest allowed. With , the constraint becomes . Corner checks: (best), , .

All coefficients in are positive and the origin is feasible ( ). Therefore the smallest value occurs at : .

Step 1: Write down the constraints: Step 2: Identify the feasible region defined by these inequalities. Step 3: Find the corner points (vertices) of the feasible region by solving the boundary equations: - Intersection of and : Substitute: - Intersection of and : Point: (feasible since ). - Intersection of and : - Intersection of and : Point: . - Intersection of and : Point: . - Check , , . Step 4: Evaluate at feasible corner points: Step 5: The minimum value of in the feasible region is at with

We are given the linear programming problem with the objective function: subject to the constraints: Step 1: Plot the constraints - The line is the boundary of the constraint . - The constraints and restrict the solution to the first quadrant. Step 2: Find the feasible region The feasible region is the area bounded by the lines: - , - , - . This forms a triangle with vertices at: - , - , - . Step 3: Evaluate the objective function at the vertices We now calculate the value of at each vertex of the feasible region: - At : - At : - At : Step 4: Conclusion The minimum value of occurs at the vertex , where . Thus, the correct answer is:

We are given the linear programming problem with the objective function: subject to the constraints: Step 1: Plot the constraints - The line is the boundary of the constraint . - The constraints and restrict the solution to the first quadrant. Step 2: Find the feasible region The feasible region is the area bounded by the line and the coordinate axes. We can find the intercepts by setting and : - When , , so the point is . - When , , so , and the point is . Thus, the feasible region is a triangle with vertices at: - , - , - . Step 3: Evaluate the objective function at the vertices We now calculate the value of at each vertex of the feasible region: - At : - At : - At : Step 4: Conclusion The maximum value of occurs at the vertex , where . Thus, the correct answer is:

We need to maximize subject to the constraints. Step 1: Identify the feasible region defined by: Step 2: Find corner points of the feasible region: - When , Corner point: and . - When , Corner point: . - Intersection point of Step 3: Evaluate at corner points: Maximum value of at .

We are given the constraints , , and . The objective function to maximize is . To maximize , we consider the boundary condition where . Substituting into , we get: To maximize , we set to its maximum value, which is 35 (as ). Substituting into : Thus, the maximum value of is 70.

We are given the constraint , with and . To maximize , we need to maximize the value of while minimizing . Since , the maximum value of occurs when . Substituting into the equation , we get: Thus, the maximum value of occurs at and , which gives: Therefore, the maximum value of is 24.