This is a typical linear programming problem where the objective is to maximize profit while satisfying constraints based on available resources and orders.

Defining Variables:

Let:

• $x$ = number of sport coats made
• $y$ = number of skirts made

Objective Function:

The objective is to maximize profit:

• The profit for each sport coat is $40.
• The profit for each skirt is $45.

So, the profit function to maximize is: $\text{Profit} = 40x + 45y$

Constraints:

1. Cotton material constraint:

   • A sport coat uses 2 yd² of cotton.
   • A skirt uses 1 yd² of cotton.
   • The tailor has 100 yd² of cotton.

   Therefore, the constraint for cotton is: $2x + y \leq 100$

2. Wool material constraint:

   • A sport coat uses 2 yd² of wool.
   • A skirt uses 3 yd² of wool.
   • The tailor has 120 yd² of wool.

   Therefore, the constraint for wool is: $2x + 3y \leq 120$

3. Minimum order constraint: The tailor must make at least 15 garments (sport coats and skirts combined). Therefore, we have: $x + y \geq 15$

4. Non-negativity constraint: The number of garments made cannot be negative: $x \geq 0 \quad \text{and} \quad y \geq 0$

Summary of the Linear Programming Problem:

Maximize: $\text{Profit} = 40x + 45y$ Subject to:

2x + y &\leq 100 \\ 2x + 3y &\leq 120 \\ x + y &\geq 15 \\ x, y &\geq 0 \end{aligned}$$ Next, we can solve this system to find the values of $$x$$ and $$y$$ that maximize the profit while satisfying the constraints. Would you like me to solve this linear programming problem step-by-step, using either graphical methods or an algebraic approach? Let me know if you'd prefer a specific method!