Let's define variables to represent the amounts of money invested in each stock:

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

We know the following:

1. The total investment is $32,000, so: $x + y = 32,000$
2. The buying limit for Stock Coff is $12,600, so: $x \leq 12,600$
3. You want to invest at least four times as much in Stock Boll as in Stock Coff, so: $y \geq 4x$
4. Stock Coff earns 4% annual interest and Stock Boll earns 6% annual interest. The total interest $I$ earned is given by: $I = 0.04x + 0.06y$

Step 1: Express $y$ in terms of $x$

From the first equation $x + y = 32,000$, we can express $y$ as: $y = 32,000 - x$

Step 2: Substitute $y$ into the inequality $y \geq 4x$

Substituting $y = 32,000 - x$ into $y \geq 4x$ gives: $32,000 - x \geq 4x$ Solving this inequality: $32,000 \geq 5x$ $x \leq \frac{32,000}{5} = 6,400$

Step 3: Determine the feasible range for $x$

We have two constraints on $x$:

• From the buying limit: $x \leq 12,600$
• From the inequality: $x \leq 6,400$

Thus, the maximum amount we can invest in Stock Coff is $x = 6,400$. The feasible range for $x$ is: $0 \leq x \leq 6,400$

Step 4: Calculate the corresponding value of $y$

When $x = 6,400$, we substitute this into $y = 32,000 - x$ to get: $y = 32,000 - 6,400 = 25,600$

Step 5: Calculate the total interest earned

Now, we compute the total interest earned using $x = 6,400$ and $y = 25,600$: $I = 0.04(6,400) + 0.06(25,600)$ $I = 256 + 1,536 = 1,792$

Final Answer:

• Amount to invest in Stock Coff: $6,400
• Amount to invest in Stock Boll: $25,600
• Maximum annual interest: $1,792

Do you need further details or clarifications? Here are five related questions to explore:

1. What happens to the total interest if the interest rates for both stocks are switched?
2. How would the solution change if you wanted to invest at least 3 times as much in Stock Boll instead of 4 times?
3. What is the effect on the maximum interest if the total amount to invest is increased to $40,000?
4. How much less annual interest would you earn if the buying limit on Stock Coff were lowered to $10,000?
5. What would be the total interest if Stock Boll's interest rate increased to 8%?

Tip: Always check your constraints carefully in optimization problems to ensure you're maximizing your return while staying within limits.