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The problem presented involves a pension fund manager who needs to invest a total of at most $35 million between U.S. Treasury bonds and mutual funds. There are constraints and fees that need to be factored in, and the goal is to maximize annual interest.

Here’s a summary of the problem:

- The fund manager plans to invest in:
  - Treasury bonds paying 6% interest
  - Mutual funds paying 8% interest
- The investment is split into:
  - At least $5 million in bonds
  - At least $15 million in mutual funds
- The total investment is capped at $35 million.
- Fees:
  - $100 per million dollars in bonds
  - $200 per million dollars in mutual funds
- The total fee should not exceed $6000.

**Objective:**
Maximize the annual interest based on the constraints.

### Let:
- $ x $ be the amount invested in Treasury bonds.
- $ y $ be the amount invested in mutual funds.

### Constraints:
1. $ x + y \leq 35 $ (Total investment is at most $35 million)
2. $ x \geq 5 $ (At least $5 million in bonds)
3. $ y \geq 15 $ (At least $15 million in mutual funds)
4. $ 100x + 200y \leq 6000 $ (Total fee constraint: $ 100x + 200y \leq 6000 $)

### Interest earned:
The total interest $ I $ can be calculated as:
\[
I = 0.06x + 0.08y
\]
The objective is to maximize this equation.

### Step-by-step approach:
1. Solve the fee constraint $ 100x + 200y \leq 6000 $.
   Dividing by 100:
   \[
   x + 2y \leq 60
   \]
2. The system of inequalities is:
   \[
   x + y \leq 35
   \]
   \[
   x + 2y \leq 60
   \]
   \[
   x \geq 5, \quad y \geq 15
   \]

Let’s solve this system to find the optimal values for $ x $ and $ y $.

### Solution:
- From $ x + 2y \leq 60 $, if $ y = 15 $, then:
  \[
  x + 2(15) \leq 60 \implies x + 30 \leq 60 \implies x \leq 30
  \]
- Also, from $ x + y \leq 35 $, if $ y = 15 $, then:
  \[
  x + 15 \leq 35 \implies x \leq 20
  \]
- Thus, $ x = 10 $ and $ y = 25 $ satisfies all the constraints.

### Maximized interest:
\[
I = 0.06(10) + 0.08(25) = 0.6 + 2 = 2.6 \, \text{million dollars}
\]

Thus, the maximum annual interest is **$2.6 million**.

Would you like more details or clarification on any part of the solution?

Here are 5 related questions you might find useful:
1. How would the solution change if the fee constraint were different?
2. What if the interest rates on Treasury bonds and mutual funds changed?
3. How do we handle more complex constraints in linear programming problems?
4. Can you explain the graphical method of solving linear programming problems?
5. What role do slack variables play in linear programming?

**Tip:** When solving linear programming problems, always check if the solution lies at a boundary point of the feasible region, as these points often give the optimal solution.

A pension fund manager decides to invest a total of at most $35 million in U.S. Treasury bonds paying 6% annual interest and in mutual funds paying 8% annual interest. The manager plans to invest at least $5 million in bonds and at least $15 million in mutual funds. The fund manager is allowed to spend no more than $6000 on fees. How much should be invested in each to maximize annual interest? What is the maximum annual interest?

Learn how to maximize annual interest by investing up to $35 million in U.S. Treasury bonds and mutual funds, while adhering to constraints on fees and minimum investments. Using linear programming, we solve this optimization problem step-by-step, ideal for advanced high school students.

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