Final Answer:
Alex should make 2 batches of cookies and 6 batches of brownies to maximize her income.
Step-by-step explanation:
To optimize income, Alex needs to balance the resources (flour and time) available while maximizing profits. Since she wants at least 2 batches of cookies due to her friends' preference, let's start with that. Two batches of cookies require 8 cups of flour (4 cups per batch x 2) and 16 minutes (8 minutes per batch x 2). With 28 cups of flour and 64 minutes in total, subtracting the resources used for cookies leaves her with 20 cups of flour and 48 minutes.
Now, let's focus on maximizing the remaining resources using brownies. Each batch of brownies requires 2 cups of flour and 6 minutes of preparation time. With 20 cups of flour and 48 minutes left, she can make 10 batches of brownies (20 cups of flour / 2 cups per batch) within the time constraint.
To determine the optimal distribution, she should calculate the profits. With cookies yielding $14 profit per batch and brownies yielding $10 profit per batch, 2 batches of cookies generate $28 ($14 x 2), while 6 batches of brownies generate $60 ($10 x 6). Combining these yields a total profit of $88 (28 + 60).
Therefore, by making 2 batches of cookies and 6 batches of brownies, Alex maximizes her profit to $88 while utilizing all the available resources within the time frame.