Problem
It takes Fena Tailoring $2 \mathrm{hr}$ of cutting and $4 \mathrm{hr}$ of sewing to make a tiered silk organza bridal dress. It takes $4 \mathrm{hr}$ of cutting and $2 \mathrm{hr}$ of sewing to make a lace sheath bridal dress. The shop has at most $16 \mathrm{hr}$ per week available for cutting and at most $20 \mathrm{hr}$ per week for sewing. The profit is $\$ 304$ on an organza dress and $\$ 221$ on a lace dress. How many of each kind of bridal dress should be made each week in order to maximize profit? What is the maximum profit? The shop should make organza dresses and lace dresses to yield a maximum profit of $\$ \square$.
Solution
Step 1 ：This problem is a linear programming problem. We need to maximize the profit function subject to the constraints of cutting and sewing time.
Step 2 ：The profit function is \(304x + 221y\), where x is the number of organza dresses and y is the number of lace dresses.
Step 3 ：The constraints are \(2x + 4y \leq 16\) (cutting time) and \(4x + 2y \leq 20\) (sewing time). We also have the non-negativity constraints \(x \geq 0\) and \(y \geq 0\).
Step 4 ：By solving this linear programming problem, we find that the optimal solution is to make 4 organza dresses and 2 lace dresses per week.
Step 5 ：This will yield a maximum profit of \(\boxed{1658}\).
