To maximize the objective function [tex]\( p = 91x + 83y \)[/tex] given the corner points [tex]\((0,0)\)[/tex], [tex]\((4,13)\)[/tex], [tex]\((11,8)\)[/tex], [tex]\((0,9)\)[/tex], and [tex]\((6,0)\)[/tex], we will evaluate the function at each point and compare the results.
1. For the point [tex]\((0, 0)\)[/tex]:
[tex]\[
p = 91 \cdot 0 + 83 \cdot 0 = 0
\][/tex]
The value of [tex]\( p \)[/tex] at [tex]\((0, 0)\)[/tex] is [tex]\( 0 \)[/tex].
2. For the point [tex]\((4, 13)\)[/tex]:
[tex]\[
p = 91 \cdot 4 + 83 \cdot 13 = 364 + 1079 = 1443
\][/tex]
The value of [tex]\( p \)[/tex] at [tex]\((4, 13)\)[/tex] is [tex]\( 1443 \)[/tex].
3. For the point [tex]\((11, 8)\)[/tex]:
[tex]\[
p = 91 \cdot 11 + 83 \cdot 8 = 1001 + 664 = 1665
\][/tex]
The value of [tex]\( p \)[/tex] at [tex]\((11, 8)\)[/tex] is [tex]\( 1665 \)[/tex].
4. For the point [tex]\((0, 9)\)[/tex]:
[tex]\[
p = 91 \cdot 0 + 83 \cdot 9 = 0 + 747 = 747
\][/tex]
The value of [tex]\( p \)[/tex] at [tex]\((0, 9)\)[/tex] is [tex]\( 747 \)[/tex].
5. For the point [tex]\((6, 0)\)[/tex]:
[tex]\[
p = 91 \cdot 6 + 83 \cdot 0 = 546 + 0 = 546
\][/tex]
The value of [tex]\( p \)[/tex] at [tex]\((6, 0)\)[/tex] is [tex]\( 546 \)[/tex].
Now, we compare the values of [tex]\( p \)[/tex] at each point:
- [tex]\( 0 \)[/tex] at [tex]\((0, 0)\)[/tex]
- [tex]\( 1443 \)[/tex] at [tex]\((4, 13)\)[/tex]
- [tex]\( 1665 \)[/tex] at [tex]\((11, 8)\)[/tex]
- [tex]\( 747 \)[/tex] at [tex]\((0, 9)\)[/tex]
- [tex]\( 546 \)[/tex] at [tex]\((6, 0)\)[/tex]
The maximum value of [tex]\( p \)[/tex] is [tex]\( 1665 \)[/tex] at the point [tex]\((11, 8)\)[/tex]. Therefore, the corner point that maximizes the objective function [tex]\( p = 91x + 83y \)[/tex] is [tex]\((11, 8)\)[/tex].
