You have $38,000 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 two times as much money on Stock Boll as Stock Coff.
If Stock Boll earns 9% 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)?

Let's define the following variables:

- Let $ x $ be the amount invested in Stock Coff.
- Let $ y $ be the amount invested in Stock Boll.

From the problem, we have the following conditions:
1. The total investment is \$38,000:
   \[
   x + y = 38,000
   \]
2. The maximum amount allowed to invest in Stock Coff is \$12,300:
   \[
   x \leq 12,300
   \]
3. You want to spend at least twice as much on Stock Boll as on Stock Coff:
   \[
   y \geq 2x
   \]
4. The interest rates are:
   - 9% for Stock Boll, i.e., 0.09 $ y $ dollars interest annually.
   - 4% for Stock Coff, i.e., 0.04 $ x $ dollars interest annually.

### Objective:
We want to **maximize** the total annual interest, which is:
\[
\text{Total interest} = 0.09y + 0.04x
\]

### Step 1: Express $ y $ in terms of $ x $
From the equation $ x + y = 38,000 $, solve for $ y $:
\[
y = 38,000 - x
\]

### Step 2: Apply the conditions
Using the condition $ y \geq 2x $, substitute $ y = 38,000 - x $:
\[
38,000 - x \geq 2x
\]
Simplifying:
\[
38,000 \geq 3x
\]
\[
x \leq \frac{38,000}{3} \approx 12,666.67
\]
Thus, $ x $ must satisfy:
\[
x \leq 12,300
\]
So the limiting condition is $ x \leq 12,300 $.

### Step 3: Maximize the interest function
The interest function is:
\[
\text{Interest} = 0.09(38,000 - x) + 0.04x
\]
Simplifying:
\[
\text{Interest} = 3,420 - 0.09x + 0.04x = 3,420 - 0.05x
\]
We want to minimize $ x $ to maximize the interest. From the earlier condition $ x \geq 12,666.67 $, the smallest value $ x $ can take is \( 12,300 \.

.





You have $38,000 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 two times as much money on Stock Boll as Stock Coff. If Stock Boll earns 9% 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)?

Solve a real-world problem involving investing $38,000 in two stocks with varying interest rates. Learn how to use algebraic inequalities and optimization to maximize your annual interest. Suitable for students in Grades 10-12.

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