You have $
3
6
,
0
0
0
you would like to invest in two different stocks, Stock Boll and Stock Coff. The buying limit on Stock Coff is $
1
2
,
3
0
0
and you want to spend at least four times as much money on Stock Boll as Stock Coff.If Stock Boll earns
6
%
annual interest and Stock Coff earns
4
%
annual interest, how much money
(
in dollars
)
should you invest in each stock to maximize your annual interest earned? What is the maximum annual interest
(
in dollars
)
?
You will earn a maximum total annual interest of $ if you invest $ in Stock Boll and $ in Stock Coff.

To solve this problem, we need to allocate the total amount of $ \$36,000 $ between Stock Boll and Stock Coff, while adhering to certain constraints. The goal is to maximize the total annual interest earned from the two stocks.

### Define Variables:
- Let $ x $ represent the amount invested in Stock Coff.
- Let $ y $ represent the amount invested in Stock Boll.

### Given Information:
1. Total available money: $ x + y = 36,000 $.
2. The limit on Stock Coff: $ x \leq 12,300 $.
3. You want to invest **at least four times** as much in Stock Boll as in Stock Coff, so: $ y \geq 4x $.
4. Stock Boll earns 6% annual interest, and Stock Coff earns 4% annual interest.
5. We need to **maximize the total annual interest** earned, which can be expressed as:
   \[
   \text{Total interest} = 0.06y + 0.04x
   \]

### Set up the system of inequalities and equations:
- $ x + y = 36,000 $ (total money constraint),
- $ x \leq 12,300 $ (buying limit on Stock Coff),
- $ y \geq 4x $ (at least four times as much invested in Boll).

### Substitution:
From $ x + y = 36,000 $, solve for $ y $:
\[
y = 36,000 - x
\]

Substitute this into the inequality $ y \geq 4x $:
\[
36,000 - x \geq 4x
\]
Solving for $ x $:
\[
36,000 \geq 5x \implies x \leq 7,200
\]

Therefore, the amount to invest in Stock Coff must satisfy:
\[
x \leq \min(12,300, 7,200) = 7,200
\]

So, $ x \leq 7,200 $, which is the limit for Stock Coff.

### Invest in Stocks:
- Invest $ x = 7,200 $ in Stock Coff.
- The amount left for Stock Boll is $ y = 36,000 - 7,200 = 28,800 $.

### Calculate Total Interest:
Now, calculate the maximum total annual interest earned:
\[
\text{Total interest} = 0.06(28,800) + 0.04(7,200)
\]
\[
= 1,728 + 288 = 2,016
\]

### Conclusion:
- You should invest **\$28,800** in Stock Boll and **\$7,200** in Stock Coff.
- The maximum total annual interest earned is **\$2,016**.

---

Would you like any more details on the calculations or steps? Here are some follow-up questions you might find useful:

1. How does the constraint $ y \geq 4x $ influence the total interest earned?
2. What happens to the maximum interest if the limit on Stock Coff changes?
3. What is the effect of changing the interest rates on the stocks?
4. Could we maximize interest without the constraint on Stock Boll's investment?
5. How would the solution change if the total investment amount were different?

**Tip:** When optimizing under constraints, always look for the intersection of limits to find the best possible solution.

You have $36,000 you would like to invest in two different stocks, Stock Boll and Stock Coff. The buying limit on Stock Coff is $12,300 and you want to spend at least four times as much money on Stock Boll as Stock Coff. If Stock Boll earns 6% annual interest and Stock Coff earns 4% annual interest, how much money should you invest in each stock to maximize your annual interest earned? What is the maximum annual interest?

Learn how to allocate $36,000 between two stocks, Stock Boll and Stock Coff, to maximize annual interest earnings. This problem involves using algebraic methods, inequality constraints, and interest formulas to determine the optimal investment distribution. Stock Boll earns 6% interest, and Stock Coff earns 4%. Suitable for Grades 10-12.

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