To solve this problem, we need to set up a system of equations and optimize the revenue, which is generated from selling two different mixes. Let's define the variables and proceed step by step:

Step 1: Define variables

Let:

• $x$ = kilograms of the first mix (half nuts and half raisins)
• $y$ = kilograms of the second mix (three-fourths nuts and one-fourth raisins)

Step 2: Set up the constraints based on the total amount of nuts and raisins available

From the problem, the total amount of chocolate-covered nuts is 138 kg, and the total amount of chocolate-covered raisins is 78 kg.

• The first mix is half nuts and half raisins. So, for $x$ kg of the first mix:

  • Nuts: $0.5x$
  • Raisins: $0.5x$

• The second mix contains three-fourths nuts and one-fourth raisins. So, for $y$ kg of the second mix:

  • Nuts: $0.75y$
  • Raisins: $0.25y$

Now, the total amounts of nuts and raisins must be used up:

1. Nuts constraint: $0.5x + 0.75y = 138$
2. Raisins constraint: $0.5x + 0.25y = 78$

Step 3: Set up the revenue function

The revenue function is based on the amount of each mix produced and sold:

• The first mix sells for $7 per kg, so the revenue from the first mix is $7x$.
• The second mix sells for $9.50 per kg, so the revenue from the second mix is $9.5y$.

Thus, the total revenue $R$ is: $R = 7x + 9.5y$

Step 4: Solve the system of equations

We need to solve the two equations:

1. $0.5x + 0.75y = 138$
2. $0.5x + 0.25y = 78$

By solving these equations, we can find $x$ and $y$, which will allow us to calculate the maximum revenue.

Let me calculate this.The company should prepare:

• 96 kg of the first mix (half nuts and half raisins)
• 120 kg of the second mix (three-fourths nuts and one-fourth raisins)

Step 5: Calculate the maximum revenue

The total revenue $R$ is given by: $R = 7x + 9.5y$ Substitute $x = 96$ and $y = 120$ into this equation to find the maximum revenue.

Let me calculate that next.The maximum revenue the company can achieve is $1,812.

Thus, for the maximum revenue, the company should prepare:

• 96 kg of the first mix
• 120 kg of the second mix
• Maximum revenue: $1,812

Would you like more details or have any questions?

Here are some related questions to consider:

1. How would the result change if the price per kg of the second mix was $10?
2. What if the company only had 100 kg of nuts available, how would that affect the optimal solution?
3. How would we approach this problem if the goal was to minimize costs instead of maximizing revenue?
4. How do changes in ingredient proportions affect overall revenue?
5. What would happen if the company wanted to maintain a 60:40 ratio of nuts to raisins in both mixes?

Tip: When solving optimization problems, setting up the correct system of equations based on constraints is key to finding the optimal solution efficiently.