A pension fund manager decides to invest a total of at most $$ \$ 45 $$ million in U.S. Treasury bonds paying $$ 3 \% $$ annual interest and in mutual funds paying $$ 5 \% $$ annual interest. She plans to invest at least $$ \$ 5 $$ million in bonds and at least $$ \$ 20 $$ 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 $$ \$ 8000 $$ on fees. How much should be invested in each to maximize annual interest? What is the maximum annual interest? The amount that should be invested in Treasury bonds is $$ \$ \square $$ million and the amount that should be invested in mutual funds is $$ \$ \square $$ million.

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Let  
$$
x = \text{investment (in millions) in Treasury bonds,}
$$
$$
y = \text{investment (in millions) in mutual funds.}
$$

The constraints are:

1. Total investment:  
   $$
   x + y \le 45.
   $$
2. Minimum investments:  
   $$
   x \ge 5 \quad \text{and} \quad y \ge 20.
   $$
3. Fees constraint:  
   $$
   100x + 200y \le 8000.
   $$
   Dividing by 100 gives:  
   $$
   x + 2y \le 80.
   $$

The annual interests are $$3\%$$ for bonds and $$5\%$$ for mutual funds. Thus the objective function (in millions) is:  
$$
\text{Interest} = 0.03x + 0.05y.
$$

### Step 1. Identify the Feasible Region

The constraints in summary are:
- $$x \ge 5$$
- $$y \ge 20$$
- $$x + y \le 45$$
- $$x + 2y \le 80$$

### Step 2. Find the Corner Points

We consider intersections of the constraints:

1. **Intersection of $$x + y = 45$$ and $$x + 2y = 80$$:**

Subtracting the first from the second:  
   $$
   (x + 2y) - (x + y) = 80 - 45 \quad \Longrightarrow \quad y = 35.
   $$
   Substitute $$y = 35$$ into $$x + y = 45$$:  
   $$
   x + 35 = 45 \quad \Longrightarrow \quad x = 10.
   $$
   So one vertex is $$(10, 35)$$.

2. **Intersection of $$x + y = 45$$ and $$y = 20$$:**

Substitute $$y = 20$$ into $$x + y = 45$$:  
   $$
   x + 20 = 45 \quad \Longrightarrow \quad x = 25.
   $$
   So another vertex is $$(25, 20)$$.

3. **Intersection of $$x + 2y = 80$$ and $$x = 5$$:**

Substitute $$x = 5$$ into $$x + 2y = 80$$:  
   $$
   5 + 2y = 80 \quad \Longrightarrow \quad 2y = 75 \quad \Longrightarrow \quad y = 37.5.
   $$
   So we obtain $$(5, 37.5)$$.

4. **Corner from the minimum constraints:**

The point $$(5, 20)$$ (when both $$x$$ and $$y$$ are at their minima) is also feasible.

### Step 3. Evaluate the Objective Function at Each Corner

We compute $$0.03x + 0.05y$$ for each candidate:

- At $$(5, 20)$$:  
  $$
  0.03(5) + 0.05(20) = 0.15 + 1.00 = 1.15.
  $$

- At $$(25, 20)$$:  
  $$
  0.03(25) + 0.05(20) = 0.75 + 1.00 = 1.75.
  $$

- At $$(5, 37.5)$$:  
  $$
  0.03(5) + 0.05(37.5) = 0.15 + 1.875 = 2.025.
  $$

- At $$(10, 35)$$:  
  $$
  0.03(10) + 0.05(35) = 0.30 + 1.75 = 2.05.
  $$

### Step 4. Identify the Optimal Investment

The maximum annual interest of $$ \$2.05 $$ million is achieved at  
$$
x = 10 \quad \text{and} \quad y = 35.
$$

### Final Answer

The amount that should be invested in Treasury bonds is $$ \$10 $$ million and the amount that should be invested in mutual funds is $$ \$35 $$ million.  
The maximum annual interest is $$ \$2.05 $$ million.
Invest $$ \$10 $$ million in Treasury bonds and $$ \$35 $$ million in mutual funds to achieve a maximum annual interest of $$ \$2.05 $$ million.

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