Maximize the objective function:

\[ z = 4x_1 + 2x_2 + 4x_3 + 6x_4 + 7x_5 \]

Subject to the constraints:

1. Board: \[ 8x_1 + 6x_2 + 8x_3 + 8x_4 + 10x_5 \leq 200 \]
2. Cable: \[ 2x_1 + 1x_2 + 2x_3 + 3x_4 + 4x_5 \leq 60 \]
3. Solder: \[ 6x_1 + 4x_2 + 4x_3 + 6x_4 + 8x_5 \leq 120 \]
4. Speaker: \[ 6x_1 + 3x_2 + 4x_3 + 6x_4 + 6x_5 \leq 100 \]

Where:
- \( x_1 \) = total acoustic
- \( x_2 \) = total canon
- \( x_3 \) = total acr
- \( x_4 \) = total black spider
- \( x_5 \) = total ashley

The question involves a linear programming problem in mathematics. The variables x1 through x5 need to be found that would maximize or minimize the objective function while complying with the given constraints: 'board, cable, solder, speaker'. The graphical or simplex method can be valuable in solving this.

Explanation:

This question presents a linear programming problem. These types of problems are often used in business and economics to maximize or minimize things like cost or profit given a set of constraints. Here, the goal is to find the values of x1, x2, x3, x4, and x5, where these variables might represent different options or items. The function to maximize or minimize (in this case, 'Max : z=4×1+2×2+4×3+6×4+7×5') is called the objective function, whereas the inequalities present the constraints (in this case, 'board, cable, solder, speaker'). The solution to the problem will be the values of the variables that maximize or minimize the objective function while still fitting within the constraints. To solve this, you may use the graphical method or simplex method in linear programming.

Learn more about Linear Programming here:

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