I'll analyze the image for the mathematical content and provide the solution step by step.

Problem Setup:

The manufacturer is solving a linear programming problem to maximize profit. The variables are:

• $x_1$: Number of racing bikes.
• $x_2$: Number of touring bikes.
• $x_3$: Number of mountain bikes.

The objective function for profit is: $z = 9x_1 + 13x_2 + 25x_3$

Constraints:

• Steel availability: $11x_1 + 13x_2 + 22x_3 \leq 28,600$
• Aluminum availability: $15x_1 + 27x_2 + 21x_3 \leq 29,400$
• Non-negativity: $x_1, x_2, x_3 \geq 0$

Let me now solve the problem systematically.

Part (a): Optimal Solution

To maximize $z$, the problem can be solved using the Simplex Method or by testing corner points of the feasible region. I will compute the solution. scipy.optimize import linprog

Coefficients of the objective function (profit per bike type)

c = [-9, -13, -25] # Negative for maximization in linprog

Coefficients for constraints (steel and aluminum)

A = [ [11, 13, 22], # Steel constraint coefficients [15, 27, 21] # Aluminum constraint coefficients ]

Right-hand side values for constraints (available materials)

b = [28600, 29400]

Bounds for each variable (non-negativity)

x0_bounds = (0, None) x1_bounds = (0, None) x2_bounds = (0, None)

Solve the linear programming problem

result = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds, x2_bounds], method='highs')

result### Part (a): Optimal Solution The optimal solution is:

• $x_1 = 0$: Racing bikes.
• $x_2 = 0$: Touring bikes.
• $x_3 = 1300$: Mountain bikes.

The company should produce 1300 mountain bikes and no racing or touring bikes.

Part (b): Maximum Profit

The maximum profit is: $z = 9(0) + 13(0) + 25(1300) = 32,500$ So, the maximum profit is $32,500.

Part (c): Material Usage

Slack variables indicate unused resources:

• Steel: Slack = $0$. All 28,600 units are used.
• Aluminum: Slack = $2100$. $29,400 - 27,300 = 2100$ units of aluminum are left unused.

Answer: (A) No, since $s_1 = 0$ and $s_2 = 2100$, there are 0 units of steel and 2100 units of aluminum left over.

Part (d): Assumptions and Real-World Issues

Answer: (A) Producing only one type of bike does not account for demand for other types.

Do you want further details, clarification, or step-by-step explanations? Here are five related questions:

1. How is the Simplex Method applied in such problems?
2. Why are slack variables important in linear programming?
3. How would a change in material availability affect the solution?
4. What if demand constraints were added to the model?
5. Can the same approach be used for non-linear problems?

Tip: When solving optimization problems, always check for constraints to ensure feasibility and interpret slack variables for leftover resources!