Find the maximum value of the function for the polygonal convex set determinded by the given system of inequalities.
8x + 2y > 36
–3x + 6y < 27
–7x + 5y > –18
f(x, y) = 9x + 5y
To find the maximum value of the function f(x, y) = 9x + 5y within the polygonal convex set determined by the given system of inequalities, we can follow these steps:
Step 1: Graph the system of inequalities.
To graph the system of inequalities, we need to plot the lines represented by each inequality and shade the appropriate regions.
The inequality 8x + 2y > 36 can be rewritten as y > -4x + 18. The line y = -4x + 18 has a y-intercept of 18 and a slope of -4, so we can plot this line as a dotted line (since it's a strict inequality) and shade the region above the line.
The inequality -3x + 6y < 27 can be rewritten as y < (1/2)x + 9. The line y = (1/2)x + 9 has a y-intercept of 9 and a slope of 1/2, so we can plot this line as a dotted line and shade the region below the line.
The inequality -7x + 5y > -18 can be rewritten as y > (7/5)x - 18/5. The line y = (7/5)x - 18/5 has a y-intercept of -18/5 and a slope of 7/5, so we can plot this line as a dotted line and shade the region above the line.
After graphing these lines and shading the appropriate regions, we will have a polygonal convex set.
Step 2: Find the vertices of the polygonal convex set.
The vertices of the polygonal convex set are the intersection points of the lines we plotted earlier.
To find the intersection points, set each pair of lines equal to each other and solve for x and y.
Setting y = -4x + 18 equal to y = (1/2)x + 9, we get:
-4x + 18 = (1/2)x + 9
Multiplying both sides by 2 to eliminate the fraction, we get:
-8x + 36 = x + 18
Combining like terms, we have:
-9x = -18
Dividing by -9, we find:
x = 2
Substituting this value of x back into either equation, we find:
y = -4(2) + 18 = 10
So one vertex is (2, 10).
Similarly, setting y = -4x + 18 equal to y = (7/5)x - 18/5, we get:
-4x + 18 = (7/5)x - 18/5
Multiplying both sides by 5 to eliminate the fraction, we get:
-20x + 90 = 7x - 18
Combining like terms, we have:
-27x = -108
Dividing by -27, we find:
x = 4
Substituting this value of x back into either equation, we find:
y = -4(4) + 18 = 2
So another vertex is (4, 2).
Step 3: Evaluate the function at each vertex.
Now that we have the vertices of the polygonal convex set, we can evaluate the function f(x, y) = 9x + 5y at each vertex.
Evaluating f(x, y) = 9x + 5y at (2, 10), we get:
f(2, 10) = 9(2) + 5(10) = 18 + 50 = 68
Evaluating f(x, y) = 9x + 5y at (4, 2), we get:
f(4, 2) = 9(4) + 5(2) = 36 + 10 = 46
Step 4: Determine the maximum value.
Comparing the values of f(x, y) at the vertices, we find that the maximum value is 68, which occurs at the vertex (2, 10).
Therefore, the maximum value of the function f(x, y) = 9x + 5y within the polygonal convex set determined by the given system of inequalities is 68.