What point in the feasible region maximizes the objective function?Constraints: x≥0y≥0−x+3≥yy≤13x+1 Objective function: C = 5x – 4y explain in simple
To find the point in the feasible region that maximizes the objective function, we first need to graph the inequalities represented by the constraints:
1. Start by graphing the line x = 0, which represents the constraint x ≥ 0. This line is a vertical line passing through the y-axis at x = 0.
2. Then graph the line y = 0, which represents the constraint y ≥ 0. This line is a horizontal line passing through the x-axis at y = 0.
3. Next, graph the line -x + 3 ≥ y. To do this, rewrite the inequality as y ≤ -x + 3 and graph the line y = -x + 3. This line has a y-intercept of 3 and a slope of -1.
4. Finally, graph the line y ≤ 13x + 1. This is an inequality in slope-intercept form, so it is easy to graph. It has a y-intercept of 1 and a slope of 13.
5. Now, shade the region that satisfies all the given constraints. This shaded region represents the feasible region, which is the set of all points that satisfy the inequalities.
Once you have the feasible region graphed, you can determine the point that maximizes the objective function. In this case, the objective function is C = 5x - 4y.
To find the maximum value, you can evaluate the objective function at each corner point of the feasible region. The corner points are the intersection points of the lines that define the feasible region.
Calculate the value of the objective function at each corner point and compare the results. The point with the highest objective function value is the solution that maximizes the objective function in the given feasible region.
The feasible region is the possible solutions to the system of inequalities. The given constraints are: x≥0, y≥0, −x+3≥y, y≤13x+1.
To find the point in the feasible region that maximizes the objective function, we need to evaluate the objective function at each corner point of the feasible region and see which one gives the highest value.
We'll start by graphing the constraints to determine the feasible region's shape and corner points.
First, let's draw the lines x=0 and y=0. These represent the x≥0 and y≥0 constraints, respectively. These lines divide the coordinate plane into four quadrants.
Next, let's draw the line −x+3=y. This represents the −x+3≥y constraint. This line has a slope of -1 and a y-intercept of 3. We can plot two points: (0, 3) and (3, 0), and then connect them with a line.
Finally, let's draw the line y=13x+1. This represents the y≤13x+1 constraint. This line has a slope of 13 and a y-intercept of 1. We can plot two points: (0, 1) and (1/13, 2), and then connect them with a line.
Now, we need to find the points where these lines intersect to identify the corner points of the feasible region. After examining the graph, we find the following corner points: (0, 3), (0, 1), (1/13, 2).
Now we evaluate the objective function, C=5x-4y, at each of these corner points.
For the point (0, 3): C = 5(0) - 4(3) = -12.
For the point (0, 1): C = 5(0) - 4(1) = -4.
For the point (1/13, 2): C = 5(1/13) - 4(2) ≈ -7.077.
From these values, we can see that the point (0, 3) gives the maximum value for the objective function. Therefore, the point (0, 3) in the feasible region maximizes the objective function.
To find the point in the feasible region that maximizes the objective function, we can follow these steps:
Step 1: Graph the feasible region.
- The constraints given are: x ≥ 0, y ≥ 0, -x + 3 ≥ y, y ≤ 13x + 1.
- Each constraint represents a line or an inequality on the x-y plane.
- Plot the lines for each constraint and shade the region that satisfies all the constraints. This shaded area is the feasible region.
Step 2: Identify the corner points of the feasible region.
- The corner points of the feasible region represent the extreme solutions.
- To find these corner points, look for the intersections of the lines formed by the constraints.
Step 3: Substitute the corner points into the objective function.
- Plug in the x and y values of each corner point into the objective function, C = 5x - 4y, to calculate the corresponding objective function value.
Step 4: Determine the maximum value of the objective function.
- Compare the objective function values obtained from each corner point.
- The corner point with the highest objective function value represents the point in the feasible region that maximizes the objective function.
Therefore, to find the point in the feasible region that maximizes the objective function, you need to follow these steps and analyze the results.