Time left 1:51: Solve the problem. Zach is planning to invest up to $\$ 45,000$ in corporate and municipal bonds. The least he will invest in corporate bonds is $\$ 8000$ and he does not want to invest more than $\$ 28,000$ in corporate, bonds. He also does not want to invest more than $\$ 28,311$ in municipal bonds. The interest is $8.2 \%$ on corporate bonds and $5.9 \%$ on municipal bonds This is simple interest for one year. What is the maximum value of his investment after one year?

Step 1 ：Let's denote the amount invested in corporate bonds as x and the amount invested in municipal bonds as y. The objective function to maximize is \(0.082x + 0.059y\).

Step 2 ：The constraints are: \(x + y \leq 45000\) (total amount invested), \(x \geq 8000\) (minimum amount in corporate bonds), \(x \leq 28000\) (maximum amount in corporate bonds), \(y \leq 28311\) (maximum amount in municipal bonds).

Step 3 ：We can solve this problem using a linear programming solver. The solver gives us the optimal solution where x = 28000 and y = 17000.

Step 4 ：Substituting these values into the objective function, we get the maximum value of the investment after one year as \(0.082*28000 + 0.059*17000 = 3299\).

Step 5 ：Final Answer: The maximum value of his investment after one year is \(\boxed{3299}\).