The solution of the system of three inequalities is given by a polygonal convex set.
8x + 2y ≥ 36
-3x + 6y ≤ 27
-7x + 5y ≥ -18
The function f(x,y) = 9x + 5y passes through this set. What values of (x,y) give f(x,y) its maximum value?
A)(-3, 2)
B)(-3, 6)
C)(9, 9)
D)(8, 2)
To find the values of (x, y) that give f(x, y) its maximum value, we need to determine the vertices of the polygonal convex set formed by the system of inequalities.
Step 1: Convert each inequality to an equation to determine the boundary lines.
8x + 2y = 36
-3x + 6y = 27
-7x + 5y = -18
Step 2: Graph each equation on the coordinate plane to determine the region enclosed by the inequalities. The region will be bounded by the intersection points of the boundary lines.
Step 3: Find the vertices of the polygonal convex set. These are the points where the boundary lines intersect.
Step 4: Substitute the coordinates of each vertex into the function f(x, y) = 9x + 5y and calculate the corresponding values.
The vertex that gives f(x, y) its maximum value is the solution.
Let's follow these steps to find the answer:
Step 1: Convert the inequalities to equations:
8x + 2y = 36
-3x + 6y = 27
-7x + 5y = -18
Step 2: Graph the equations on the coordinate plane:
The graph is a triangle with vertices at (-3, 6), (3, 6), and (2, 2).
Step 3: Find the vertices of the polygonal convex set:
The vertices are (-3, 6), (3, 6), and (2, 2).
Step 4: Substitute the coordinates of each vertex into the function f(x, y) = 9x + 5y:
f(-3, 6) = 9(-3) + 5(6) = -27 + 30 = 3
f(3, 6) = 9(3) + 5(6) = 27 + 30 = 57
f(2, 2) = 9(2) + 5(2) = 18 + 10 = 28
Therefore, the maximum value of f(x, y) corresponds to the vertex (3, 6).
The answer is B) (-3, 6).