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Let's break down the problem:

### Problem:
- Total investment: **At most $25 million**
- Treasury bonds (4% annual interest) with a minimum of **$5 million** invested.
- Mutual funds (7% annual interest) with a minimum of **$10 million** invested.
- Bonds fee: **$100 per million dollars**.
- Mutual funds fee: **$200 per million dollars**.
- Total fees: **Cannot exceed $4000**.

The goal is to find:
1. How much to invest in Treasury bonds and mutual funds to **maximize the interest**.
2. The maximum **annual interest** generated.

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

### Constraints:
1. $ x + y \leq 25 $ (Total investment cannot exceed $25 million).
2. $ x \geq 5 $ (At least $5 million in Treasury bonds).
3. $ y \geq 10 $ (At least $10 million in mutual funds).
4. Fee constraint: $ 100x + 200y \leq 4000 $ (Total fee cannot exceed $4000).

### Objective:
Maximize the total annual interest, which is given by:
\[
I = 0.04x + 0.07y
\]
subject to the above constraints.

### Fee Constraint Simplification:
The fee constraint $ 100x + 200y \leq 4000 $ simplifies to:
\[
x + 2y \leq 40
\]

Now we have the following system of inequalities:
1. $ x + y \leq 25 $
2. $ x + 2y \leq 40 $
3. $ x \geq 5 $
4. $ y \geq 10 $

### Solution:
We'll solve this system by testing the boundaries of these inequalities and calculating the interest at each point.

The feasible region for the constraints is formed by the intersection of the inequalities. Solving these will give us the optimal values of $ x $ and $ y $.

Let me calculate it for you.The optimal solution is to invest:
- **$10 million** in Treasury bonds.
- **$15 million** in mutual funds.

This allocation results in a **maximum annual interest** of **$1.45 million**.

Would you like further details or have any questions?

Here are 5 related questions to expand on this:
1. How would the result change if the fee budget was increased or decreased?
2. What happens if the interest rates of Treasury bonds and mutual funds change?
3. How can the solution be found graphically for a better understanding?
4. How would changes in the minimum investment amounts affect the optimal solution?
5. What if there were other types of investments added to the problem?

### Tip:
Linear programming problems like this are often solved using boundary values because the optimal solution usually lies at the intersection of the constraints.

A pension fund manager decides to invest a total of at most $25 million in U.S. Treasury bonds paying 4% annual interest and in mutual funds paying 7% annual interest. He plans to invest at least $5 million in bonds and at least $10 million in mutual funds. Bonds have an initial fee of $100 per million dollars, while the fee for mutual funds is $200 per million. The fund manager is allowed to spend no more than $4000 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 in Treasury bonds and mutual funds with given constraints. This problem involves linear programming, interest rate optimization, and fee constraints. Suitable for students in Grades 11-12 or college-level courses.

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