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The problem provided involves maximizing the income from two different investments: municipal bonds and Treasury bills. Here’s a breakdown of the given information and the steps to solve it:

- You have **$18,000** to invest.
- At least **$4,000** should go into **municipal bonds**, which yield **5% interest**.
- No more than **$2,000** should be invested in **Treasury bills**, which yield **8% interest**.
- You are asked to determine how much should be placed in each investment to **maximize interest**.

### Step 1: Define Variables
Let:
- $ x $ be the amount of money invested in municipal bonds.
- $ y $ be the amount of money invested in Treasury bills.

### Step 2: Set Up Constraints
1. The total amount invested is $18,000, so:
   \[
   x + y = 18,000
   \]
2. At least $4,000 must be invested in municipal bonds:
   \[
   x \geq 4,000
   \]
3. No more than $2,000 can be invested in Treasury bills:
   \[
   y \leq 2,000
   \]

### Step 3: Objective Function (Maximize Interest)
The objective is to maximize the interest earned, which is given by:
\[
\text{Interest} = 0.05x + 0.08y
\]
We aim to maximize this function subject to the above constraints.

### Step 4: Substituting Total Investment Equation
From the equation $ x + y = 18,000 $, we can express $ y $ in terms of $ x $:
\[
y = 18,000 - x
\]
Substitute this into the interest formula:
\[
\text{Interest} = 0.05x + 0.08(18,000 - x)
\]
Simplify:
\[
\text{Interest} = 0.05x + 1,440 - 0.08x
\]
\[
\text{Interest} = 1,440 - 0.03x
\]
### Step 5: Maximizing the Interest
Since $ 1,440 - 0.03x $ decreases as $ x $ increases, the interest is maximized when $ x $ is **as small as possible**. According to the constraint $ x \geq 4,000 $, the smallest value $ x $ can take is $4,000.

Thus, the maximum interest occurs when:
- $ x = 4,000 $
- $ y = 18,000 - 4,000 = 14,000 $

However, this violates the constraint $ y \leq 2,000 $. Therefore, we need to adjust.

### Step 6: Apply the Constraint on $ y $
To satisfy $ y \leq 2,000 $, the maximum $ y $ can be is **$2,000**. This implies:
\[
y = 2,000
\]
Then, $ x = 18,000 - 2,000 = 16,000 $.

### Step 7: Calculate Maximum Interest
Now, substitute $ x = 16,000 $ and $ y = 2,000 $ into the interest formula:
\[
\text{Interest} = 0.05(16,000) + 0.08(2,000)
\]
\[
\text{Interest} = 800 + 160 = 960
\]

Thus, the **maximum income is $960**, and it occurs when $ x = 16,000 $ and $ y = 2,000 $.

### Final Answer:
- Maximum income: **$960**
- Maximum income at: $ (16,000, 2,000) $

Let me know if you would like more details on any part of the solution or if you have any questions! Here are 5 related questions to explore:

1. How does the interest change if the yield on municipal bonds increases to 6%?
2. What would happen if there were no limit on the amount invested in Treasury bills?
3. How would the solution change if the total available money was $20,000 instead of $18,000?
4. What if you were required to invest at least $5,000 in municipal bonds?
5. How would the solution change if simple interest was replaced by compound interest?

**Tip:** Always check the constraints of your problem after solving to ensure your solution is valid.

On your birthday your rich uncle gave you $18000. You would like to invest at least $4000 of the money in municipal bonds yielding 5% and no more than $2000 in Treasury bills yielding 8%. How much should be placed in each investment in order to maximize the interest earned in one year? Assume simple interest applies.

Solve an optimization problem involving $18,000 invested in municipal bonds and Treasury bills. Learn how to maximize interest with constraints such as investing at least $4,000 in municipal bonds (5% interest) and no more than $2,000 in Treasury bills (8% interest). Includes a step-by-step solution.

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