To determine which ordered pair maximizes the objective function [tex]\( P = 3x + 8y \)[/tex] among the given pairs, we will evaluate the objective function at each point.
Given ordered pairs:
1. [tex]\((0, 0)\)[/tex]
2. [tex]\((2, 7)\)[/tex]
3. [tex]\((5, 6)\)[/tex]
4. [tex]\((8, 1)\)[/tex]
Let's evaluate the objective function [tex]\( P = 3x + 8y \)[/tex] for each of these pairs:
1. For the pair [tex]\((0, 0)\)[/tex]:
[tex]\[ P = 3(0) + 8(0) = 0 + 0 = 0 \][/tex]
2. For the pair [tex]\((2, 7)\)[/tex]:
[tex]\[ P = 3(2) + 8(7) = 6 + 56 = 62 \][/tex]
3. For the pair [tex]\((5, 6)\)[/tex]:
[tex]\[ P = 3(5) + 8(6) = 15 + 48 = 63 \][/tex]
4. For the pair [tex]\((8, 1)\)[/tex]:
[tex]\[ P = 3(8) + 8(1) = 24 + 8 = 32 \][/tex]
Next, we compare the values we obtained:
- [tex]\( P(0, 0) = 0 \)[/tex]
- [tex]\( P(2, 7) = 62 \)[/tex]
- [tex]\( P(5, 6) = 63 \)[/tex]
- [tex]\( P(8, 1) = 32 \)[/tex]
Among these values, the maximum value of [tex]\( P \)[/tex] is 63, which corresponds to the ordered pair [tex]\((5, 6)\)[/tex]. Therefore, the ordered pair [tex]\((5, 6)\)[/tex] maximizes the objective function [tex]\( P = 3x + 8y \)[/tex].
