On your birthday your great aunt gave you $\$ 20000$. You would like to invest at least $\$ 4000$ of the money in municipal bonds yielding $3 \%$ and no more than $\$ 2000$ in Treasury bills yielding $4 \%$. How much should be placed in each investment in order to maximize the interest earned in one year? Assume simple interest applies. Let $x$ represent the amount of money in municipal bonds and $y$ represent the amount of money in Treasury bills. Answer

Step 1 ：First, we need to understand the constraints of the problem. We have a total of $20000, and we want to invest at least $4000 in municipal bonds and no more than $2000 in Treasury bills. So, we have $4000 \leq x \leq 20000$ and $0 \leq y \leq 2000$.

Step 2 ：Next, we need to set up the objective function, which is the total interest earned in one year. The interest from the municipal bonds is $0.03x$ and the interest from the Treasury bills is $0.04y$. So, the total interest is $0.03x + 0.04y$.

Step 3 ：Since we want to maximize the total interest, we need to find the maximum value of the objective function under the constraints. The maximum value of $0.03x$ is achieved when $x = 20000$, and the maximum value of $0.04y$ is achieved when $y = 2000$.

Step 4 ：So, to maximize the total interest, we should invest $20000 in municipal bonds and $2000 in Treasury bills. However, this would exceed our total amount of $20000. So, we need to adjust our investment.

Step 5 ：Since the interest rate of Treasury bills is higher than that of municipal bonds, we should invest as much as possible in Treasury bills. So, we invest $2000 in Treasury bills.

Step 6 ：Then, we invest the remaining $20000 - $2000 = $18000 in municipal bonds.

Step 7 ：Finally, we check our solution. We have $18000 in municipal bonds and $2000 in Treasury bills, which meets the constraints. The total interest is $0.03 \times 18000 + 0.04 \times 2000 = \boxed{660}$, which is the maximum possible under the constraints.