Identify the feasible region for the following set of const…
Identify the feasible region for the following set of constraints: 0.54A + 0.25B greater than or equal to 30 1A + 5B greater than or equal to 250 0.25A + 0.5B less than or equal to 50 A, B greater than or equal to 0
Answer
To identify the feasible region for the given set of constraints, we need to plot the inequalities on a graph and find the overlapping region.
Let’s begin by graphing the first inequality: 0.54A + 0.25B ≥ 30. To do this, we can start by finding two points on the line corresponding to this inequality. By setting A = 0, we find B = 120, and by setting B = 0, we find A = 55. Plotting these points on a graph, we can draw a line connecting them.
Next, let’s graph the second inequality: 1A + 5B ≥ 250. By setting A = 0, we find B = 50, and by setting B = 0, we find A = 250. Plotting these points and connecting them with a line, we obtain a second line on the graph.
Lastly, we graph the third inequality: 0.25A + 0.5B ≤ 50. By setting A = 0, we find B = 100, and by setting B = 0, we find A = 200. Similar to the previous step, we plot these points and connect them to form a line on the graph.
By examining the graph, we can see that the feasible region corresponds to the area where all three lines overlap or intersect. This is the region where all the constraints are satisfied simultaneously.
Now, let’s interpret the meaning of the feasible region in the context of the problem. In this case, we have two variables, A and B, with non-negative constraints (greater than or equal to 0). The feasible region represents the combinations of A and B that satisfy all the given constraints.
For example, any point within the feasible region, such as (60, 20), would satisfy all the inequalities. This means that if we plug in the values of A=60 and B=20 into the original equations, we would obtain results that are greater than or equal to 30 for the first inequality, greater than or equal to 250 for the second inequality, and less than or equal to 50 for the third inequality.
On the other hand, any point outside the feasible region, such as (-10, 80), would not satisfy all the constraints. For this point, plugging in A=-10 and B=80 into the original equations would yield results that violate one or more of the inequalities.
In conclusion, the feasible region for the given set of constraints is the overlapping region on the graph where all three lines intersect. This region represents the combinations of A and B that satisfy the inequalities, resulting in a feasible solution.