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Use a graphing calculator or other technology to solve the following linear programming problem.

Maximize [tex]z = 37x_1 + 34x_2 + 36x_3 + 30x_4 + 35x_5[/tex]

Subject to:

[tex]
\begin{align*}
16x_1 + 19x_2 + 23x_3 + 15x_4 + 21x_5 & \leq 35,000 \\
15x_1 + 10x_2 + 19x_3 + 23x_4 + 10x_5 & \leq 29,000 \\
9x_1 + 16x_2 + 14x_3 + 12x_4 + 11x_5 & \leq 23,000 \\
18x_1 + 20x_2 + 15x_3 + 17x_4 + 19x_5 & \leq 31,000
\end{align*}
[/tex]

With [tex]x_1 \geq 0[/tex], [tex]x_2 \geq 0[/tex], [tex]x_3 \geq 0[/tex], [tex]x_4 \geq 0[/tex], [tex]x_5 \geq 0[/tex].

The maximum is [tex]\square[/tex] when [tex]x_1 = \square[/tex], [tex]x_2 = \square[/tex], [tex]x_3 = \square[/tex], [tex]x_4 = \square[/tex], [tex]x_5 = \square[/tex], [tex]s_1 = \square[/tex], [tex]s_2 = \square[/tex], [tex]s_3 = \square[/tex], and [tex]s_4 = \square[/tex].

(Round to the nearest hundredth as needed.)

Answer :

To solve this linear programming problem, we want to maximize the objective function while satisfying a set of constraints. Let's walk through the process step-by-step:

### Objective Function:
The objective function we want to maximize is:
[tex]\[ z = 37x_1 + 34x_2 + 36x_3 + 30x_4 + 35x_5 \][/tex]

### Constraints:
We have four inequality constraints and non-negativity constraints for the variables:
1. [tex]\( 16x_1 + 19x_2 + 23x_3 + 15x_4 + 21x_5 \leq 35,000 \)[/tex]
2. [tex]\( 15x_1 + 10x_2 + 19x_3 + 23x_4 + 10x_5 \leq 29,000 \)[/tex]
3. [tex]\( 9x_1 + 16x_2 + 14x_3 + 12x_4 + 11x_5 \leq 23,000 \)[/tex]
4. [tex]\( 18x_1 + 20x_2 + 15x_3 + 17x_4 + 19x_5 \leq 31,000 \)[/tex]

All variables must be non-negative:
[tex]\[ x_1 \geq 0, \, x_2 \geq 0, \, x_3 \geq 0, \, x_4 \geq 0, \, x_5 \geq 0 \][/tex]

### Solution:
After analyzing the problem, the values for the variables that maximize the objective function given the constraints are:

- [tex]\( x_1 = 930.48 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex]
- [tex]\( x_3 = 679.14 \)[/tex]
- [tex]\( x_4 = 0 \)[/tex]
- [tex]\( x_5 = 213.90 \)[/tex]

The maximum value of the objective function is:
[tex]\[ z = 66,363.64 \][/tex]

### Summary:
- The maximum value of [tex]\( z \)[/tex] occurs when [tex]\( x_1 = 930.48 \)[/tex], [tex]\( x_2 = 0 \)[/tex], [tex]\( x_3 = 679.14 \)[/tex], [tex]\( x_4 = 0 \)[/tex], and [tex]\( x_5 = 213.90 \)[/tex].
- The value of the maximum objective function is [tex]\( 66,363.64 \)[/tex] (rounded to two decimal places).

By keeping the constraints in mind and optimizing the allocation of resources (the variables [tex]\( x_1 \)[/tex] to [tex]\( x_5 \)[/tex]), this solution achieves the highest possible outcome for the given problem.

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