- A bakery produces two types of cookies: chocolate chip and caramel. The bakery anticipates daily demand for a minimum of 80 caramelized and 120 chocolate chip cookies. Due to a lack raw of materials and labor, the bakery can produce 120 caramel cookies and 140 chocolate chip cookies daily. For the bakery to be viable, it must sell a minimum of 240 cookies each day. Every chocolate chip cookie served generates $\$ 0.75$ in profit, whereas each caramel cookie generates $\$ 0.88$. Determine the number of chocolate chip and caramel cookies that the bakery must produce each day to maximize profit using linear programming.

Step 1 ：Define the variables: Let \(x\) be the number of chocolate chip cookies and \(y\) be the number of caramel cookies.

Step 2 ：Identify the constraints: \(y \geq 80\), \(x \geq 120\), \(y \leq 120\), \(x \leq 140\), and \(x + y \geq 240\).

Step 3 ：Identify the objective function: The profit, which is \(0.75x + 0.88y\), needs to be maximized.

Step 4 ：Convert the problem into a minimization problem by multiplying the profit by -1. The new objective function is \(-0.75x - 0.88y\).

Step 5 ：Solve the problem using a linear programming solver. The solver returns that the optimal solution is \(x = 120\) and \(y = 120\).

Step 6 ：Substitute \(x = 120\) and \(y = 120\) back into the profit function to find the maximum profit: \(0.75(120) + 0.88(120) = \$195.6\).

Step 7 ：Final Answer: The bakery should produce \(\boxed{120}\) chocolate chip cookies and \(\boxed{120}\) caramel cookies each day to maximize profit. The maximum profit is \(\boxed{\$195.6}\).