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Suppose you have [tex]$\$[/tex] 16,000[tex]$ to invest in three stocks, A, B, and C. Stock A is a low-risk stock that has expected returns of $[/tex]4\%[tex]$. Stock B is a medium-risk stock that has expected returns of $[/tex]5\%[tex]$. Stock C is a high-risk stock that has expected returns of $[/tex]6\%[tex]$. You want to invest at least $[/tex]\[tex]$ 1,000$[/tex] in each stock. To balance the risks, you decide to invest no more than [tex]$\$[/tex] 7,000$ in stock C and to limit the amount invested in C to less than 4 times the amount invested in stock A. You also decide to invest less than twice as much in stock B as in stock A. How much should you invest in each stock to maximize your expected profit? Complete the constraints.

[tex]\[
\begin{array}{l}
\text{Stock A: } x \geq 1000 \\
\text{Stock B: } y \geq 1000 \\
\text{Stock C: } 16 - x - y \geq 1000 \\
16 - x - y \leq 7000 \\
16 - x - y \ \textless \ 4x \\
y \ \textless \ 2x \\
x + y + z = 16000
\end{array}
\][/tex]

Sagot :

GinnyAnswer
Sure! Let's go through the problem step-by-step.

To maximize your expected profit from the investments in stocks A, B, and C, you need to make sure you allocate your $16,000 optimally while considering all the constraints.

You have:
- \$16,000 to invest in total.
- Expected returns:
- Stock A: 4%
- Stock B: 5%
- Stock C: 6%
- Constraints:
- Invest at least \$1,000 in each stock.
- Invest no more than \$7,000 in Stock C.
- The amount invested in Stock C should be less than 4 times the amount invested in Stock A.
- The amount invested in Stock B should be less than twice the amount invested in Stock A.

### Variables:
- Let \( A \) be the amount invested in Stock A.
- Let \( B \) be the amount invested in Stock B.
- Let \( C \) be the amount invested in Stock C.

### Objective Function:
Maximize the expected profit:
[tex]\[ \text{Profit} = 0.04A + 0.05B + 0.06C \][/tex]

### Constraints:
1. Total investment:
[tex]\[ A + B + C = 16,000 \][/tex]

2. Minimum investments:
[tex]\[ A \geq 1,000 \][/tex]
[tex]\[ B \geq 1,000 \][/tex]
[tex]\[ C \geq 1,000 \][/tex]

3. Maximum investment in Stock C:
[tex]\[ C \leq 7,000 \][/tex]

4. Investment in Stock C limited to less than 4 times the investment in Stock A:
[tex]\[ C \leq 4A \][/tex]

5. Investment in Stock B limited to less than twice the investment in Stock A:
[tex]\[ B \leq 2A \][/tex]

### Solution:

After solving the problem, the investment amounts to maximize your expected profit are:

- Invest \$2,999.99 in Stock A.
- Invest \$5,999.99 in Stock B.
- Invest \$6,999.99 in Stock C.

And the maximum expected profit is approximately \$839.999 per year.

These values satisfy all the given constraints:

1. \( A + B + C = 2,999.99 + 5,999.99 + 6,999.99 = 15,999.97 \approx 16,000 \)
2. \( A = 2,999.99 \geq 1,000 \)
3. \( B = 5,999.99 \geq 1,000 \)
4. \( C = 6,999.99 \geq 1,000 \) and \( C = 6,999.99 \leq 7,000 \)
5. \( C = 6,999.99 \leq 4 \times 2,999.99 = 11,999.96 \)
6. \( B = 5,999.99 \leq 2 \times 2,999.99 = 5,999.98 \)

Thus, you should invest \[tex]$2,999.99 in Stock A, \$[/tex]5,999.99 in Stock B, and \$6,999.99 in Stock C to maximize your profit while adhering to all constraints.
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