Sara is aspiring to venture into the chocolate manufacturing business in California. She envisions creating delectable chocolate

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Question:

Sara is aspiring to venture into the chocolate manufacturing business in California. She envisions creating delectable chocolate treats and selling them locally.

Given her limited initial capital and lack of experience in large-scale production, Sara plans to start on a small scale. She intends to invest $15,000 from her savings for purchasing raw materials. Sara is considering applying for a bank loan to further support her venture. To ensure a successful start, Sara needs to make crucial decisions regarding the allocation of funds between the two main types of chocolate treats she plans to produce: premium dark chocolates and assorted milk chocolates.

In the first year, each box of premium dark chocolates requires $2 worth of raw materials, while each box of assorted milk chocolates uses $1.5 worth of raw materials. In the second year, the costs per box are $1.8 and $1.3, respectively.

Sara estimates that in the first year, a box of premium dark chocolates will be sold for $10, and a box of assorted milk chocolates will be sold for $8.5. In the second year, the selling prices are expected to increase to $12 for premium dark chocolates and $9 for assorted milk chocolates.

Additionally, Sara needs to allocate a portion of her budget for advertising to boost sales. The local chocolate lovers' association has provided guidance on the relationship between advertising expenditure and expected demand. For each dollar spent on advertising premium dark chocolates in the first year, an estimated demand for eight boxes will be generated. In the second year, for each dollar spent on advertising, a demand for ten boxes is expected.

Similarly, for assorted milk chocolates, the association suggests that in the first year, up to ten boxes can be sold for each advertising dollar spent, and in the second year, up to twelve boxes.

The initial funds for the advertising will come from the $15,000 savings. Assume that the cash earned from chocolate sales in the first year is available in the second year.

Considering the limited advertising budget, Sara aims to maximize her profit while ensuring a balance between the sales of premium dark chocolates and assorted milk chocolates. She wants to maintain the sales ratio within 45% to 70% for premium dark chocolates.

Question: Sara seeks your assistance in determining the optimal allocation of funds for purchasing raw materials and advertising. Develop a mathematical model to help Sara make informed decisions, ensuring maximum profit over the two-year period. Solve the model to provide Sara with detailed recommendations for her chocolate manufacturing business.