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