Let's break down the problem step by step.

Part (a):

The company has 138 kg of chocolate-covered nuts and 78 kg of chocolate-covered raisins. The products will be sold in two different mixes:

• Mix 1 contains half nuts and half raisins, and sells for $7 per kg.
• Mix 2 contains $\frac{3}{4}$ nuts and $\frac{1}{4}$ raisins, and sells for $9.50 per kg.

The company needs to decide how many kilograms of each mix to prepare in order to maximize revenue.

Let:

• $x_1$ be the kilograms of Mix 1.
• $x_2$ be the kilograms of Mix 2.

Each mix uses up the nuts and raisins according to its recipe, and the constraints are:

• Nuts constraint: $\frac{1}{2}x_1 + \frac{3}{4}x_2 \leq 138$ (from the total amount of nuts available).
• Raisins constraint: $\frac{1}{2}x_1 + \frac{1}{4}x_2 \leq 78$ (from the total amount of raisins available).

The objective is to maximize the revenue, which is given by: $\text{Revenue} = 7x_1 + 9.5x_2$

We solve this linear programming problem to determine the values of $x_1$ and $x_2$.

Part (b):

Now, the price of Mix 2 is raised to $11 per kg. The revenue equation becomes: $\text{Revenue} = 7x_1 + 11x_2$ The same constraints apply. We solve this revised problem to find the new optimal amounts of Mix 1 and Mix 2, and the maximum revenue.

I'll calculate the values using the appropriate method.### Results:

Part (a):

• The company should prepare 96 kg of Mix 1 and 120 kg of Mix 2.
• The maximum revenue will be $1812.

Part (b):

• After the price increase for Mix 2, the company should prepare 0 kg of Mix 1 and 184 kg of Mix 2.
• The maximum revenue will be $2024.

Would you like any more details or clarification?

Follow-up Questions:

1. How are the constraints for the nuts and raisins derived?
2. Why does the price change in Mix 2 affect the preparation amounts so drastically?
3. How would the solution change if the price of Mix 1 increased?
4. What other techniques could be used to solve this optimization problem?
5. Can you explain how the linear programming method finds the optimal solution?

Tip:

When dealing with multiple products and limited resources, adjusting prices can dramatically shift production strategies to maximize revenue.