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 \( \$ 10 \) million and the amount that should be invested in mutual funds is \( \$ 35 \) million. The maximum annual interest is \( \$ \square \).

Let \[ x = \text{amount (in millions) invested in bonds}, \quad y = \text{amount (in millions) invested in mutual funds}. \] **Constraints:** 1. Total investment: \[ x + y \le 45 \] 2. Minimum investment in bonds: \[ x \ge 5 \] 3. Minimum investment in mutual funds: \[ y \ge 20 \] 4. Fees: Bonds incur a fee of \(\$100\) per million, and mutual funds incur \(\$200\) per million. The fee constraint is: \[ 100x + 200y \le 8000. \] Dividing by 100, we get: \[ x + 2y \le 80. \] **Objective Function:** The annual interest earned is: \[ 0.03x + 0.05y, \] and we wish to maximize this value. **Given Candidate:** It is provided that the optimal solution involves: \[ x = 10 \quad \text{and} \quad y = 35. \] **Verification of Constraints:** 1. Total investment: \[ x + y = 10 + 35 = 45 \quad (\text{meets } x+y \le 45) \] 2. Minimum in bonds: \[ 10 \ge 5 \] 3. Minimum in mutual funds: \[ 35 \ge 20 \] 4. Fee constraint: \[ x + 2y = 10 + 2(35) = 10 + 70 = 80 \quad (\text{meets } x+2y \le 80) \] **Calculation of the Maximum Annual Interest:** Substitute \(x = 10\) and \(y = 35\) into the objective function: \[ 0.03(10) + 0.05(35) = 0.3 + 1.75 = 2.05. \] Thus, the maximum annual interest is \(\$2.05\) million. **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.