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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.

A candy company has 138 kg of chocolate-covered nuts and 78 kg of chocolate-covered raisins to be sold as two different mixes. One mix will contain half nuts and half raisins and will sell for $7 per kg. The other mix will contain ¾ nuts and ¼ raisins and will sell for $9.50 per kg. How many kilograms of each mix should the company prepare for maximum revenue?

This problem explores how to maximize the revenue of a candy company using linear programming techniques. Given constraints on the amount of nuts and raisins, learn how to allocate resources between two mixes to achieve maximum profit, considering price changes.

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