To determine the minimum and maximum values of the objective function [tex]\(F = 8x + 5y\)[/tex] for the given vertices of the feasible region [tex]\((0, 100), (0, 80), (80, 60), (80, 0), (120, 0)\)[/tex], we need to evaluate the objective function at each vertex:
1. For the vertex [tex]\((0, 100)\)[/tex]:
[tex]\[
F = 8(0) + 5(100) = 0 + 500 = 500
\][/tex]
2. For the vertex [tex]\((0, 80)\)[/tex]:
[tex]\[
F = 8(0) + 5(80) = 0 + 400 = 400
\][/tex]
3. For the vertex [tex]\((80, 60)\)[/tex]:
[tex]\[
F = 8(80) + 5(60) = 640 + 300 = 940
\][/tex]
4. For the vertex [tex]\((80, 0)\)[/tex]:
[tex]\[
F = 8(80) + 5(0) = 640 + 0 = 640
\][/tex]
5. For the vertex [tex]\((120, 0)\)[/tex]:
[tex]\[
F = 8(120) + 5(0) = 960 + 0 = 960
\][/tex]
Now we have the values of the objective function at each vertex:
- At [tex]\((0, 100)\)[/tex], [tex]\(F = 500\)[/tex]
- At [tex]\((0, 80)\)[/tex], [tex]\(F = 400\)[/tex]
- At [tex]\((80, 60)\)[/tex], [tex]\(F = 940\)[/tex]
- At [tex]\((80, 0)\)[/tex], [tex]\(F = 640\)[/tex]
- At [tex]\((120, 0)\)[/tex], [tex]\(F = 960\)[/tex]
The minimum value of [tex]\(F\)[/tex] is 400, and the maximum value of [tex]\(F\)[/tex] is 960.
Therefore:
Minimum: [tex]\(400\)[/tex]
Maximum: [tex]\(960\)[/tex]
