Pick the corner point below that maximizes the following objective function:

[tex]p = 91x + 83y[/tex]

A. [tex]\((0, 0)\)[/tex]

B. [tex]\((4, 13)\)[/tex]

C. [tex]\((11, 8)\)[/tex]

D. [tex]\((0, 9)\)[/tex]

E. [tex]\((6, 0)\)[/tex]



Answer :

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].